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Regulators of cubic fields
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Numbers
$D$
$k$ 
$R_K$
-23
1:
0.2811995743229618465120507640678782997920232257440664626757303343180384530621208913229877074754940548
comment: $a^3-a^2+1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a$; LMFDB 3.1.23.1; $\varepsilon=\rho$ is the plastic number, $\rho^3=\rho+1$, so $R_K=\log\rho$, OEIS A202539
-31
1:
0.3822450858400356413293584991848573937594164224201954300292839383616548905505831820170135085159009127
comment: $a^3+a-1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+1$; LMFDB 3.1.31.1
-44
1:
0.6093778634360062315368033711683986954285392793128541477762892366225152051203195768397057073419585497
comment: $a^3-a^2+a+1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a+1$; LMFDB 3.1.44.1
49
1:
0.5254546821225723883388260454483245095410550460117947850676492724203110593549975930800866136421569825
comment: $a^3-a^2-2a+1=0$; $C_3$, conductor $7$; $h_K=1$; $\varepsilon_1=a^2-a-1$, $\varepsilon_2=a$; LMFDB 3.3.49.1; $K=\mathbb{Q}(\zeta_7)^+$, the maximal real subfield of the seventh cyclotomic field
-59
1:
0.7909857206389212540018693272530749506345743338628627440008206646227339683565767634104040550579281526
comment: $a^3+2a-1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2$; LMFDB 3.1.59.1
-76
1:
1.018591819777906590354224673681164915994520640696143243699845880252487185614047098135736284964771120
comment: $a^3-2a-2=0$; $S_3$; $h_K=1$; $\varepsilon=a+1$; LMFDB 3.1.76.1
81
1:
0.8492874506461925286447620049374467362147951797113786103684368382038257910024099980313368867859083713
comment: $a^3-3a-1=0$; $C_3$, conductor $9$; $h_K=1$; $\varepsilon_1=a^2-a-2$, $\varepsilon_2=a$; LMFDB 3.3.81.1; $K=\mathbb{Q}(\zeta_9)^+$
-83
1:
1.040692599440395485092298348257543952124706481079311857055228174249750493669938489963909575258479366
comment: $a^3-a^2+a-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+1$; LMFDB 3.1.83.1
-87
1:
0.9348448455461977860783140525196923570024867308486850838450888916855682445322844141181203081483658377
comment: $a^3-a^2+2a+1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a+2$; LMFDB 3.1.87.1
-104
1:
1.576082953228427907613973872231958709224898184994129471659614891815891676860753407533919546786020611
comment: $a^3-a-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+1$; LMFDB 3.1.104.1
-107
1:
1.256056721471871168960710247617748821905169854854499370432317584769676384596156592415197699421516640
comment: $a^3-a^2+3a-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+3$; LMFDB 3.1.107.1
-108
1:
1.347377348329384100918187891445653046283062273320657774182479121695539186758470703258250239376552912
comment: $a^3-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+1$; LMFDB 3.1.108.1; $K=\mathbb{Q}(\sqrt[3]{2})$, a pure cubic field
-116
1:
1.717529795978247094697849970184609277100477610714292129213754784204892643092581498375347306294336458
comment: $a^3-a^2-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+1$; LMFDB 3.1.116.1
-135
1:
1.132628263567543830737020997508998744255967095970016646938389487405273855958177643930606130884554463
comment: $a^3+3a-1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+3$; LMFDB 3.1.135.1
-139
1:
1.663554735366906795745511969430855390521187326141453162213095156979603503057755102550545108910891965
comment: $a^3-a^2+a+2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+3$; LMFDB 3.1.139.1
-140
1:
1.473670882843464308820966821901786809074156569066843270445770100309451784539655145548790705749169951
comment: $a^3+2a-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+3$; LMFDB 3.1.140.1
148
1:
1.662336520767896261901574137885589416799186131341565774524116085355581814131131481096344693082488132
comment: $a^3-a^2-3a+1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a^2-2$; LMFDB 3.3.148.1
-152
1:
2.130645585956807526556212020204723305944940169106235610526427984766444545704303979597970977169871932
comment: $a^3-a^2-2a-2=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+1$; LMFDB 3.1.152.1
169
1:
1.365049867594382595635083804801990945164514047163020195518192691645742478211845751185315302555855630
comment: $a^3-a^2-4a-1=0$; $C_3$, conductor $13$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=a$; LMFDB 3.3.169.1; the cubic subfield of $\mathbb{Q}(\zeta_{13})$
-172
1:
1.881952909620126364335992167636974448499700375840097884733831870957008008934319425430749110345304757
comment: $a^3-a^2-a+3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+2$; LMFDB 3.1.172.1
-175
1:
1.288506512723927955829860552773154596049884266430652085709472292207694361537493317237207991158425927
comment: $a^3-a^2+2a-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2$; LMFDB 3.1.175.1
-199
1:
1.336657776982638754417090365997950270960670090595124845634190421217836095233027817837168408928017739
comment: $a^3-a^2+4a-1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a+4$; LMFDB 3.1.199.1
-200
1:
2.604399407588468247848287079401802085671043528739281006350695993912416663727871209577562117656743424
comment: $a^3-a^2+2a+2=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2-5a+9$; LMFDB 3.1.200.1
-204
1:
2.354643678754182673642775694739537696786742241462452761423862112321264387379433335742729636691811119
comment: $a^3-a^2+a-3=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+a+4$; LMFDB 3.1.204.1
-211
1:
2.237632872796399210299437255458887043006445841588879716630856601624955393677196127969523491939981648
comment: $a^3-2a-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2a+2$; LMFDB 3.1.211.1
-212
1:
2.712756209080331359122366032821732972087115428372283849286050067703800348868642174857080841690383484
comment: $a^3-a^2+4a-2=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2-2a+15$; LMFDB 3.1.212.1
-216
1:
3.023577078711816253653549138191635827272985778634680305567839617974676108515148726797857501933122714
comment: $a^3+3a-2=0$; $S_3$; $h_K=1$; $\varepsilon=5a^2+3a+17$; LMFDB 3.1.216.1
229
1:
2.355454590844564506512740955367661703561955611786719328265973866256185619262257629086680304233038265
comment: $a^3-4a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a+2$; LMFDB 3.3.229.1
-231
1:
1.745490135151363625619789170413661696374584803543660710375344404508522302746391435767496263401807935
comment: $a^3-a^2+3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+2$; LMFDB 3.1.231.1
-239
1:
2.096542023629994762716904054459903728772772535117260290663632692764304719757463052507214599026884640
comment: $a^3-a-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2a+2$; LMFDB 3.1.239.1
-243
1:
2.524681404706315896966265280407903468389442199467792772240703072357836633309794216389785460581832835
comment: $a^3-3=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+3a+4$; LMFDB 3.1.243.1; $K=\mathbb{Q}(\sqrt[3]{3})$, a pure cubic field
-244
1:
3.302675510490706626436021212742047552755174691911770906020229860954346446249293495152652899720271757
comment: $a^3+a-6=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2+5a+11$; LMFDB 3.1.244.1
-247
1:
1.544532289488692543446880183826438657357523062340835577119077840821048813977900316661469285746471647
comment: $a^3+a-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+2$; LMFDB 3.1.247.1
-255
1:
1.992944834450688900046543016965398049751698024664971721687266266498017888856686493707467113230831296
comment: $a^3-a^2-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+2$; LMFDB 3.1.255.1
257
1:
1.974593870780707163855999286452564513946017897094632552615123173882602263010666353170916101455832076
comment: $a^3-a^2-4a+3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-4$, $\varepsilon_2=a^2+a-2$; LMFDB 3.3.257.1
-268
1:
2.521233336354852858569921719655885565199516946824605650640976767910248809831070095140778616543385047
comment: $a^3-a^2-3a+5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-3a+3$; LMFDB 3.1.268.1
-283
1:
1.401342327308812661850449007055876367036976238101096245011464371726157379243146620104485627027558603
comment: $a^3+4a-1=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+4$; LMFDB 3.1.283.1
-300
1:
3.148549575662643079320928983110947039922435856767688187028726136346710390287047621193829358621520556
comment: $a^3-a^2-3a-3=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+3a+2$; LMFDB 3.1.300.1; $K=\mathbb{Q}(\sqrt[3]{10})$, a pure cubic field
-307
1:
2.958148258703396555095383326639852209367832351477238842304988335309038273358317802331394158091042867
comment: $a^3-a^2+3a+2=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2-6a+15$; LMFDB 3.1.307.1
316
1:
3.913458492113946900963460794549574246982412499835021399042132651763091089861807055636075090654634547
comment: $a^3-a^2-4a+2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-3$, $\varepsilon_2=a^2-a-5$; LMFDB 3.3.316.1
321
1:
2.569259680131251187516092156259545853130273780888996325197158175083743516978916579419292113571257615
comment: $a^3-a^2-4a+1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a^2-2a-1$; LMFDB 3.3.321.1
-324
1:
4.047641726449878163420166293472572184347632455544951643782949452565306458724136692279554984347830154
comment: $a^3-3a-4=0$; $S_3$; $h_K=1$; $\varepsilon=5a^2+11a+9$; LMFDB 3.1.324.1
-327
1:
2.198591639203520469748321679107224299690012939474424363795467093348883734028493476124350961961331377
comment: $a^3-a^2-2a-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+1$; LMFDB 3.1.327.1
-331
1:
1.502852833316712366561195525097579483904381230438074197131119278404371606688136677179541911549567183
comment: $a^3-a^2+3a-4=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+3$; LMFDB 3.1.331.1
-335
1:
1.455733027607560849409037041009659321985124509849095145358412669423687070020941327820762451060884414
comment: $a^3-a^2+4a+1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a+4$; LMFDB 3.1.335.1
-339
1:
3.546269237220498743516378767910913968509614098907424376204292191619099893261183392624754540164436025
comment: $a^3-a^2-a+4=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2-10a+11$; LMFDB 3.1.339.1
-351
1:
1.702275358343855726564509388201348823402437070489279652641019933523166897469005133773308450278056815
comment: $a^3+3a-3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+4$; LMFDB 3.1.351.1
-356
1:
3.755210035416743142747676388590284273783447677953655899718490416448242085318832728198543382419608390
comment: $a^3-a^2+a+7=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2-10a+19$; LMFDB 3.1.356.1
361
1:
1.952156696507314682108419837066598131143836390368129400538935552637400152116798229896482018343735667
comment: $a^3-a^2-6a+7=0$; $C_3$, conductor $19$; $h_K=1$; $\varepsilon_1=a^2+a-4$, $\varepsilon_2=a^2-5$; LMFDB 3.3.361.1; the cubic subfield of $\mathbb{Q}(\zeta_{19})$
-364
1:
2.936173922746069741004106426729677814329862928048058312842091561034727326661744164599555300143947232
comment: $a^3+4a-2=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+2a+17$; LMFDB 3.1.364.1
-367
1:
1.856197373975265926284287815635946096027060670507822135480444293746853929083416541135056434194776063
comment: $a^3-a^2+2a+3=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+4$; LMFDB 3.1.367.1
-379
1:
3.272625280091877940351972504368951234350368142982164142731784609723887953248191873983399369032473739
comment: $a^3-a^2+a-4=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+3a+9$; LMFDB 3.1.379.1
404
1:
3.759879121892829257485362167765778013091730114245238141452240125724450719672025484757485098888032765
comment: $a^3-a^2-5a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a^2+a-1$; LMFDB 3.3.404.1
-411
1:
4.028530943387016653458918141607715716315961081036388273609522118836067166563855410435842063563218101
comment: $a^3-a^2+5a-2=0$; $S_3$; $h_K=1$; $\varepsilon=12a^2-7a+57$; LMFDB 3.1.411.1
-419
1:
3.344765598265333646904695372490823113475911156604830004139702440965294871095008281747637050239593633
comment: $a^3-4a-5=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+5a+4$; LMFDB 3.1.419.1
-424
1:
4.858548533691782471918315938457826223719082914156885537654035545988690908995128904711490558875248281
comment: $a^3-a^2+8=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(21a^2-57a+98)$; LMFDB 3.1.424.1
-431
1:
6.154682255628670014502263113176224056904469965205617426162532446189369569618122888771265130150021024
comment: $a^3-a-8=0$; $S_3$; $h_K=1$; $\varepsilon=36a^2+78a+133$; LMFDB 3.1.431.1
-436
1:
4.948105715103293813111796081026536855330749020502274333700317828471115742831111907849717033544880330
comment: $a^3+a-4=0$; $S_3$; $h_K=1$; $\varepsilon=21a^2+29a+61$; LMFDB 3.1.436.1
-439
1:
2.430086614967457048824527830071822483453174174344278053499443139735970378199030402747433966867892125
comment: $a^3-a^2-2a+5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-3a+3$; LMFDB 3.1.439.1
-440
1:
4.534330537944632582937743821117352006842247770358680533936331692684276686267491960070619364093480811
comment: $a^3+2a-8=0$; $S_3$; $h_K=1$; $\varepsilon=9a^2+15a+43$; LMFDB 3.1.440.1
-451
1:
3.575734133392213388694838557709261617859383550914266544452341447063271465477710131631646350781583704
comment: $a^3-a^2-5a+8=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2-7a+7$; LMFDB 3.1.451.1
-459
1:
3.669443433748797700752844372517898831233577829034349249392372816560516337699192789952657651223292687
comment: $a^3+3a-8=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+6a+21$; LMFDB 3.1.459.1
-460
1:
3.670732895680941283284778441804070409288037917660422618816552451144911120161260226896697815453304592
comment: $a^3-a^2+5a-3=0$; $S_3$; $h_K=1$; $\varepsilon=8a^2-3a+38$; LMFDB 3.1.460.1
469
1:
3.852826615623859990265924528404243257013813539765855322032854718606816105735000324497908782563727151
comment: $a^3-a^2-5a+4=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-5$, $\varepsilon_2=a^2-2a-1$; LMFDB 3.3.469.1
-472
1:
5.380470340975060361372507274860836952791343890084878478441095275639905338139172791987383217235634685
comment: $a^3-5a-6=0$; $S_3$; $h_K=1$; $\varepsilon=13a^2+35a+29$; LMFDB 3.1.472.1
473
1:
2.843229399572870671206573564483091011826266606511196907335000738450888967840241131902782815698501149
comment: $a^3-5a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a+2$; LMFDB 3.3.473.1
-484
1:
5.302685659788826254330551979508375698145554295582663822995759406180438128444241855877555483696139683
comment: $a^3-a^2+4a+2=0$; $S_3$; $h_K=1$; $\varepsilon=37a^2-53a+171$; LMFDB 3.1.484.1
-491
1:
1.891064461062121356984126605356521465761457895312706465492160946477103689651963887498102514260166981
comment: $a^3-a^2+a+4=0$; $S_3$; $h_K=2$; $\varepsilon=a^2-2a+3$; LMFDB 3.1.491.1
-492
1:
4.420724090681489505538524984512560793748412663080365544619094489115122604180237141049688315631413893
comment: $a^3-a^2+3a+3=0$; $S_3$; $h_K=1$; $\varepsilon=14a^2-24a+59$; LMFDB 3.1.492.1
-499
1:
3.873804012284158466190991740269355939390700167960313588662444298843591736253257347550273820509566054
comment: $a^3+4a-3=0$; $S_3$; $h_K=1$; $\varepsilon=9a^2+6a+40$; LMFDB 3.1.499.1
-503
1:
7.027346793361095523685257094823939311180249000462249511519263727057985480184013939726056044310123360
comment: $a^3-a^2+2a+8=0$; $S_3$; $h_K=1$; $\varepsilon=101a^2-247a+559$; LMFDB 3.1.503.1
-515
1:
3.645812326912024404572972658806222114208885019165363869238672274194508222625404476033510011805454140
comment: $a^3-a^2-a-4=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+5a+7$; LMFDB 3.1.515.1
-516
1:
6.385077789864091070630980631079599237515857430934219990821232643989459885905979288498799693934767632
comment: $a^3-a^2+a-9=0$; $S_3$; $h_K=1$; $\varepsilon=49a^2+63a+193$; LMFDB 3.1.516.1
-519
1:
2.680579997055260240708284077627910496576317231770165172653380702359729389418973783794862575631242333
comment: $a^3-a^2-4a+7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-3a+3$; LMFDB 3.1.519.1
-524
1:
3.422374279554084899891573270933374986890345830185401218185558820089240810033630909841877770360253308
comment: $a^3-a^2+3a-5=0$; $S_3$; $h_K=1$; $\varepsilon=5a^2+2a+18$; LMFDB 3.1.524.1
-527
1:
1.617282528441683165736577166198543746069177897920407437367180034179234101292729496390019951274955085
comment: $a^3+5a-1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+5$; LMFDB 3.1.527.1
-543
1:
3.012906736903902250691295200253918984364306220702712212390742834245691339952826089905398724874004928
comment: $a^3-a^2+2a-5=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2+2a+9$; LMFDB 3.1.543.1
-547
1:
4.367037727875609247222629091614394863722881839317307644722911853459099424582439696310741360872522642
comment: $a^3-a^2-3a-4=0$; $S_3$; $h_K=1$; $\varepsilon=6a^2+10a+9$; LMFDB 3.1.547.1
-563
1:
1.736743998256103676254575959197239812170875923435937127815658346047649482137384796283941146844373314
comment: $a^3-a^2+5a-4=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+5$; LMFDB 3.1.563.1
564
1:
5.402747909983130918628412974593374547321879856412337934503787627029663576126226993974265788661054777
comment: $a^3-a^2-5a+3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+2$, $\varepsilon_2=2a^2-a-10$; LMFDB 3.3.564.1
-567
1:
2.464020392579788582963935815283152285012730305835006601648168892669563638459483041568609980998319975
comment: $a^3-3a-5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2a+2$; LMFDB 3.1.567.1
568
1:
6.087390907136765781237577770293426235806551172201706727672162016431927274476173032214561910277686084
comment: $a^3-a^2-6a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-5$, $\varepsilon_2=2a^2-4a-7$; LMFDB 3.3.568.1
-588
1:
1.654004199425936196155377142508451735150824447370078103295421012802737230072356462219146370157244040
comment: $a^3-a^2+5a+1=0$; $S_3$; $h_K=3$; $\varepsilon=a^2-a+5$; LMFDB 3.1.588.1; $K=\mathbb{Q}(\sqrt[3]{28})$, a pure cubic field
-620
1:
3.552913589190603617743254099522493366255818902522354164948078061920930110320741286187502368522948976
comment: $a^3-a^2-5a-5=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+4a+3$; LMFDB 3.1.620.1
621
1:
5.399702803080467333358959242042538282542861637722191612242508999542092816180277965527714334682470026
comment: $a^3-6a-3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+2$, $\varepsilon_2=2a+1$; LMFDB 3.3.621.1
-628
1:
6.493760709473242994593491646208586612315321987566037669219578171518016607852116404239514498426742925
comment: $a^3-a^2+4a+8=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(123a^2-271a+818)$; LMFDB 3.1.628.1
-643
1:
2.358610814594547666529559022188919091472070294909591689452421716148717342029078393910223319290764540
comment: $a^3-2a-5=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+2a+2$; LMFDB 3.1.643.1
-648
1:
2.233548287742422922378014169180468936672404527353088527582676685420212118765293481917720556695118289
comment: $a^3-3a-10=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(a^2+3a+4)$; LMFDB 3.1.648.1
-652
1:
4.320499068331619671428471142427797931578257598872955675080778716809499447208253966953113496809501292
comment: $a^3-a^2+7a+5=0$; $S_3$; $h_K=1$; $\varepsilon=8a^2-13a+64$; LMFDB 3.1.652.1
-655
1:
2.905619805204161902268241117113811642416562004107935048365450277862752018847447049446713238622078828
comment: $a^3-a^2+5=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2-5a+7$; LMFDB 3.1.655.1
-671
1:
2.345084108917704241961293732728940289900255857416027866766172417388747179929907772679283850976814972
comment: $a^3-a-5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2a+3$; LMFDB 3.1.671.1
-675
1:
4.811986539509132215684264676959213526580346092307265669435741401526612058042011804094522907475831718
comment: $a^3-5=0$; $S_3$; $h_K=1$; $\varepsilon=14a^2+24a+41$; LMFDB 3.1.675.1; $K=\mathbb{Q}(\sqrt[3]{5})$, a pure cubic field
-676
1:
2.186081359264027636661464091450333552196450980890982145363851228941506456860595661354572939187143658
comment: $a^3-a^2-4a+12=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(a^2-3a+4)$; LMFDB 3.1.676.1
-679
1:
3.443255909573521722441102799940796129846887834885320137234839560557969393764609845676286964383367269
comment: $a^3+a-5=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+6a+13$; LMFDB 3.1.679.1
-680
1:
6.070711071023635329255548374173069960411167887330310203865695067478233581637782313932885008686956298
comment: $a^3-a^2-6a+10=0$; $S_3$; $h_K=1$; $\varepsilon=21a^2-77a+79$; LMFDB 3.1.680.1
-687
1:
3.454848252560781525354503731917765630336253056937405292604289860089032510419642547399688522170256875
comment: $a^3-a^2+4a+3=0$; $S_3$; $h_K=1$; $\varepsilon=5a^2-8a+25$; LMFDB 3.1.687.1
-695
1:
2.151209987590383875443492717422719898271039053161450359453853068262638619734520529138405704888047498
comment: $a^3-a^2-5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+2$; LMFDB 3.1.695.1
-696
1:
7.809934413960878609420502609909114520299845590626477090343809874418850557185042268282502017112895100
comment: $a^3-a^2-2a+6=0$; $S_3$; $h_K=1$; $\varepsilon=207a^2-589a+673$; LMFDB 3.1.696.1
697
1:
2.711849659875934602566354435137030369636519586939988619391511152727610508556271690876143841449734865
comment: $a^3-7a-5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-6$, $\varepsilon_2=a+2$; LMFDB 3.3.697.1
-707
1:
4.186891984342934599849519709098419917982360584219633730495186417540385107128992118626800140277887202
comment: $a^3+2a-5=0$; $S_3$; $h_K=1$; $\varepsilon=9a^2+12a+34$; LMFDB 3.1.707.1
-716
1:
6.405362306307440546212784521815618550713031364532119891074299188154856416979188860610478628198214861
comment: $a^3-4a-6=0$; $S_3$; $h_K=1$; $\varepsilon=40a^2+101a+95$; LMFDB 3.1.716.1
-728
1:
6.052150076039488017260715700718163867688884458688161104552351063356396001638714838312654249731179761
comment: $a^3-a^2+6a-2=0$; $S_3$; $h_K=1$; $\varepsilon=75a^2-49a+433$; LMFDB 3.1.728.1
-731
1:
2.012678692448036242224523461819822583371154241798649503352850509238253362985034871356164457232405536
comment: $a^3-a^2+3a+4=0$; $S_3$; $h_K=2$; $\varepsilon=a^2-2a+5$; LMFDB 3.1.731.1
733
1:
5.309084110215107881804123364684946062213179517668619405798395290789937324932919363494719243349164321
comment: $a^3-a^2-7a+8=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-1$, $\varepsilon_2=2a-5$; LMFDB 3.3.733.1
-743
1:
4.555646206892134187825260954006856553347915548187725529085939545032842541773090685308217687951913057
comment: $a^3+5a-3=0$; $S_3$; $h_K=1$; $\varepsilon=16a^2+9a+85$; LMFDB 3.1.743.1
-744
1:
8.294299296528893422159189362642840038700689420776844251441526472760049913971402631484034139114443054
comment: $a^3-a^2-6a-6=0$; $S_3$; $h_K=1$; $\varepsilon=193a^2+451a+347$; LMFDB 3.1.744.1
-748
1:
4.531790413972073586042584098829298393659129993621013301558700607687216666042887183150051865419061958
comment: $a^3-a^2+a+5=0$; $S_3$; $h_K=1$; $\varepsilon=11a^2-25a+43$; LMFDB 3.1.748.1
-751
1:
1.767883792375448932002710840963385668035834070262486572490023446911682260034968773463051471571384896
comment: $a^3-a^2+6a-1=0$; $S_3$; $h_K=2$; $\varepsilon=a^2-a+6$; LMFDB 3.1.751.1
-755
1:
4.904010788499686574716756879796109155140837838545877448097775336761898769021119943217382886696631617
comment: $a^3-a^2+5a+2=0$; $S_3$; $h_K=1$; $\varepsilon=22a^2-30a+121$; LMFDB 3.1.755.1
-756
1:
7.107459681784164433477352136633260854069268699325032865467075759592505646869697509047408865789218560
comment: $a^3-6a-12=0$; $S_3$; $h_K=1$; $\varepsilon=52a^2+163a+199$; LMFDB 3.1.756.1
756
1:
5.691718404606611612993611186640669050961998302968104745766727160262303987124710858909249148393787623
comment: $a^3-6a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-5$, $\varepsilon_2=2a^2-a-11$; LMFDB 3.3.756.1
-759
1:
3.137460793131831268621991255829245387294978137915257928088955730266933279099914985003877303585799362
comment: $a^3-a^2+6a-3=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2-2a+23$; LMFDB 3.1.759.1
761
1:
3.526280520445694103091800278375944809007759915339945521165234481619932983696144793972892120381398990
comment: $a^3-a^2-6a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a+2$; LMFDB 3.3.761.1
-771
1:
6.139708014307201120984564143823598878149646327406316390241554342428475166574567252164488107250444970
comment: $a^3-a^2+3a-6=0$; $S_3$; $h_K=1$; $\varepsilon=65a^2+36a+251$; LMFDB 3.1.771.1
-780
1:
6.159207121543629845020409435342558062192664657890821773838174997766207401561266150192849282470774486
comment: $a^3-a^2-a-5=0$; $S_3$; $h_K=1$; $\varepsilon=44a^2+59a+94$; LMFDB 3.1.780.1
785
1:
4.098266025831314232189673795860628497789352497161121796966785274734112283838732385425366901953806855
comment: $a^3-a^2-6a+5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-1$, $\varepsilon_2=a^2-a-4$; LMFDB 3.3.785.1
788
1:
5.986546006426557725207159358800764806024406723993638396919519363274929721084959370016772450623809786
comment: $a^3-a^2-7a-3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+2$, $\varepsilon_2=2a+1$; LMFDB 3.3.788.1
-804
1:
8.571486833428211179213801810077489634275214668838682162109157478126259221636718039343327846912642754
comment: $a^3-a^2+4a-6=0$; $S_3$; $h_K=1$; $\varepsilon=784a^2+270a+3499$; LMFDB 3.1.804.1
-808
1:
7.625086901937183085908044514700380406672818954294997619294063123992946983081457598124124198633121052
comment: $a^3-a^2+2a-6=0$; $S_3$; $h_K=1$; $\varepsilon=259a^2+201a+875$; LMFDB 3.1.808.1
-812
1:
3.844219809060239712004498940932831557233612666387393900126273691255598964246335755747504261773375926
comment: $a^3-a^2-7a-7=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+5a+4$; LMFDB 3.1.812.1
-815
1:
5.063595226630906586826407505630641950654134464312989292469166074548359497162181004533582318500631268
comment: $a^3-7a-9=0$; $S_3$; $h_K=1$; $\varepsilon=7a^2+22a+20$; LMFDB 3.1.815.1
-823
1:
2.933322292584497507612029801143764711292582022101276223078439352136569571799077422369675027955828116
comment: $a^3-5a-7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+3a+3$; LMFDB 3.1.823.1
-835
1:
6.131424928391340841957502822859015979764708736045455180756975060392579591536814854198618710129215629
comment: $a^3-a^2-a+6=0$; $S_3$; $h_K=1$; $\varepsilon=42a^2-113a+149$; LMFDB 3.1.835.1
837
1:
6.801369525869562043381198512298272287096968963334385440383197047802539593209561642909591891582210546
comment: $a^3-6a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=2a^2+6a+3$; LMFDB 3.3.837.1
-839
1:
3.205890261021762771286712037728562329125245970113092996381211603500059345148338141472287029472294127
comment: $a^3-a^2-2a-5=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+3a+4$; LMFDB 3.1.839.1
-843
1:
6.735861815393197442798074319570196659621188314836303638178471369858899599076554526008641841617923845
comment: $a^3-a^2-2a+12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(95a^2-309a+506)$; LMFDB 3.1.843.1
-856
1:
7.582234644140657139328170539321084025129587720638953589473262290563300498651609920715492997645324862
comment: $a^3-a^2+a+11=0$; $S_3$; $h_K=1$; $\varepsilon=136a^2-382a+827$; LMFDB 3.1.856.1
-863
1:
2.021220268830687105609365682329328903341251661168967710578892181841928606449889556695673966248008067
comment: $a^3-a^2+2a+5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+4$; LMFDB 3.1.863.1
-867
1:
6.879298644727446243361290413050750400308974277135878440275196566685531508621857369813382692581922688
comment: $a^3-a^2+6a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(175a^2+119a+1250)$; LMFDB 3.1.867.1; $K=\mathbb{Q}(\sqrt[3]{17})$, a pure cubic field
-876
1:
5.947960009967695550995818224497231969921909855548032029483852098021874812124741412709802392516992549
comment: $a^3-a^2-a-11=0$; $S_3$; $h_K=1$; $\varepsilon=23a^2+41a+91$; LMFDB 3.1.876.1
-883
1:
5.897078405855122047552168383385522449449360699983883113402274374586434823785346170380298866280416396
comment: $a^3-a^2+2a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(53a^2+71a+272)$; LMFDB 3.1.883.1
-888
1:
9.641409189100985835095019623262242617876709041820593260312898029905130710694534905474750624650141567
comment: $a^3-a^2+9a+3=0$; $S_3$; $h_K=1$; $\varepsilon=1548a^2-2041a+14582$; LMFDB 3.1.888.1
-891
1:
1.796336422784922951353994194600882312995279229252574342865350823325733023401488325706017592950049173
comment: $a^3+6a-1=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+6$; LMFDB 3.1.891.1
892
1:
8.323199362987780866573853995566509937053728430109655677578059529532226311414086752505031496489236420
comment: $a^3-a^2-8a+10=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2+a-3$, $\varepsilon_2=3a^2+5a-11$; LMFDB 3.3.892.1
-907
1:
5.996278795937134952614934301690787032940413457465774518847246831691099886885295481173461709509405716
comment: $a^3-a^2-7a+12=0$; $S_3$; $h_K=1$; $\varepsilon=17a^2-66a+71$; LMFDB 3.1.907.1
-908
1:
7.715572118552969686250252540245052886531811322108032376214789255093100025915085500700059267019118419
comment: $a^3-4a-12=0$; $S_3$; $h_K=1$; $\varepsilon=109a^2+312a+457$; LMFDB 3.1.908.1
-931
1:
1.846928841527999371104114209683539537741082265550480226652153576951416203118496177750587907798628841
comment: $a^3-a^2+5a-6=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+5$; LMFDB 3.1.931.1
-932
1:
6.859666650995542023860130054929152606830601335885441752909770239526549071235109498704851372194907997
comment: $a^3+5a-4=0$; $S_3$; $h_K=1$; $\varepsilon=145a^2+105a+801$; LMFDB 3.1.932.1
-940
1:
5.837857945453859205617495238428175625234579355036302180772682413493167601008301171376224733353984334
comment: $a^3-2a-6=0$; $S_3$; $h_K=1$; $\varepsilon=28a^2+61a+77$; LMFDB 3.1.940.1
940
1:
8.908186819339298518377790837115300019824433194310512838790213026747978990668611931985108264642340834
comment: $a^3-7a-4=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2+a-3$, $\varepsilon_2=a^2-a-1$; LMFDB 3.3.940.1
-948
1:
10.05865173091829655807809380373184617727286794808743936682176147507564551396394859787057434844807403
comment: $a^3-a^2+6=0$; $S_3$; $h_K=1$; $\varepsilon=2297a^2-5829a+8963$; LMFDB 3.1.948.1
-959
1:
1.822567985913560054089172003105319908008389722553103332798010948197590695052753033998511141358136090
comment: $a^3-a^2+6a+1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a+6$; LMFDB 3.1.959.1
961
1:
12.19578191258585233051463551067581177963268483969094533405185144307579884115522857216223959719442407
comment: $a^3-a^2-10a+8=0$; $C_3$, conductor $31$; $h_K=1$; $\varepsilon_1=a^2+3a-3$, $\varepsilon_2=2a^2-19$; LMFDB 3.3.961.1; the cubic subfield of $\mathbb{Q}(\zeta_{31})$
-964
1:
7.987875971815869331416958198807020576370814931306985141399111373517839774987793116845559580433779066
comment: $a^3-2a-12=0$; $S_3$; $h_K=1$; $\varepsilon=164a^2+423a+763$; LMFDB 3.1.964.1
-971
1:
8.428147867553734694729635709083475784721000765772257585793499812278846204148447650295961019828659565
comment: $a^3-a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(545a^2+1327a+2686)$; LMFDB 3.1.971.1
-972
1:
5.105835485554791888818939596312060989769866109981849542161302993034397884548181371707325231651929360
comment: $a^3-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(21a^2+48a+110)$; LMFDB 3.1.972.1; $K=\mathbb{Q}(\sqrt[3]{12})$, a pure cubic field
-972
2:
5.789932142319291516526504589619737345928762523177042760871772558944202048608960111257644941051037639
comment: $a^3-6=0$; $S_3$; $h_K=1$; $\varepsilon=33a^2+60a+109$; LMFDB 3.1.972.2; $K=\mathbb{Q}(\sqrt[3]{6})$, a pure cubic field
-980
1:
2.534472002003700430472980890473416673717810403014019652039813512205061598096131596755818602821998844
comment: $a^3-a^2+5a-13=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+a+7$; LMFDB 3.1.980.1
-983
1:
1.905787669349791718117443323434253705882712162311501857738287076673182829279845359719573456569043374
comment: $a^3-a^2+6a-5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+6$; LMFDB 3.1.983.1
-984
1:
9.979198411261783762101317283203888805075001895152925179437111054275399239570736124547793321062554488
comment: $a^3-a^2-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(2675a^2+4483a+11996)$; LMFDB 3.1.984.1
985
1:
3.724173660672038137385695333099656333093993847871938775403059504260827017091200760921005498184965083
comment: $a^3-a^2-6a+1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a+2$; LMFDB 3.3.985.1
993
1:
5.554643955296285283940861769138395388936970230355240262654038476347803964725085299004946610139000596
comment: $a^3-a^2-6a+3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-5$, $\varepsilon_2=a^2-5$; LMFDB 3.3.993.1
-996
1:
10.02866556670071204563182370381549742319436006994697386773716065202869651724509669573232070199700315
comment: $a^3-a^2-6=0$; $S_3$; $h_K=1$; $\varepsilon=2194a^2+2674a+5933$; LMFDB 3.1.996.1
-999
1:
12.51057866118002752814236485387739672927314216524921912755640265181549232827199427683414876311479492
comment: $a^3+3a-12=0$; $S_3$; $h_K=1$; $\varepsilon=20289a^2+37715a+130975$; LMFDB 3.1.999.1
-1004
1:
7.736745016202390683802295742662186001699135893925990491575203340779341206737415242950538926693585571
comment: $a^3+2a-6=0$; $S_3$; $h_K=1$; $\varepsilon=274a^2+399a+1129$; LMFDB 3.1.1004.1
-1007
1:
2.601686134111300811060689435489565617687984179477908236505447477158660755527864377360688729181893535
comment: $a^3-a^2-2a+7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-3a+4$; LMFDB 3.1.1007.1
-1011
1:
7.095089128800829698860373777811786488349884982169680045628733890422656341107600734714633811197158356
comment: $a^3-a^2-5a-6=0$; $S_3$; $h_K=1$; $\varepsilon=64a^2+139a+121$; LMFDB 3.1.1011.1
1016
1:
10.12892707264502741049531365884741619269385198547486103273593695799894382470421753809334014875640189
comment: $a^3-a^2-6a+2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-7$, $\varepsilon_2=a^2+a-11$; LMFDB 3.3.1016.1
-1036
1:
5.231537445137206595463164393209650085335326029122062373908766156999108343628778045476292754828079015
comment: $a^3+4a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(29a^2+50a+202)$; LMFDB 3.1.1036.1
-1048
1:
8.949885986174116062491468886601427285902865603016560268649973351317503670353293519910416424687273410
comment: $a^3-a^2+8a-12=0$; $S_3$; $h_K=1$; $\varepsilon=695a^2+279a+5951$; LMFDB 3.1.1048.1
-1055
1:
12.07609493969672770613230232104554792527332655860107829150584280901840078517217633392812394020979964
comment: $a^3-a^2-6a+16=0$; $S_3$; $h_K=1$; $\varepsilon=6889a^2-27031a+37699$; LMFDB 3.1.1055.1
-1059
1:
7.478761636888623296622191482631829155292199064401632180936645128477445190265031080233064157561615901
comment: $a^3-a^2+a+6=0$; $S_3$; $h_K=1$; $\varepsilon=185a^2-442a+799$; LMFDB 3.1.1059.1
-1067
1:
5.612701647562327344088085066709552273446842368814011565926273415043271082429535135384208049753874130
comment: $a^3-4a-7=0$; $S_3$; $h_K=1$; $\varepsilon=17a^2+44a+46$; LMFDB 3.1.1067.1
-1068
1:
7.169380239751982989686652721923427819301974592864377502952561610881309426298603430199540418473138191
comment: $a^3-a^2-a+13=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(147a^2-468a+875)$; LMFDB 3.1.1068.1
-1075
1:
6.583326954055794370915201248962235633725948321172249459439080054666598414780389401276759788225992659
comment: $a^3-a^2+2a+12=0$; $S_3$; $h_K=1$; $\varepsilon=49a^2-135a+335$; LMFDB 3.1.1075.1
1076
1:
6.932326757238295437613411653207360870803136672847631632522756762967408626889128180981899277633676503
comment: $a^3-8a-6=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=3a+7$; LMFDB 3.3.1076.1
-1080
1:
9.889641444050390369314965072381907861698733440374478857007422734454272974584343736705454425594077126
comment: $a^3+3a-6=0$; $S_3$; $h_K=1$; $\varepsilon=2473a^2+3185a+11521$; LMFDB 3.1.1080.1
-1083
1:
2.629072959873913406745676564822284152612448106151805247349903809618158035116338156752638054206108610
comment: $a^3-a^2-6a-12=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(a^2+3a+4)$; LMFDB 3.1.1083.1; $K=\mathbb{Q}(\sqrt[3]{19})$, a pure cubic field
-1087
1:
3.426656794944274000977639335766557038749483190035271043325536542873173681023327413838435232368258687
comment: $a^3-a^2+4a-7=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+2a+19$; LMFDB 3.1.1087.1
-1096
1:
8.778784918547765393984566017883488518815335545426657009952011196537151957911823471846723256417554993
comment: $a^3-a^2+a-13=0$; $S_3$; $h_K=1$; $\varepsilon=413a^2+650a+2086$; LMFDB 3.1.1096.1
-1099
1:
3.181579439839739528575322951333806047546251407288763313614250568478594561314727643070675883333916383
comment: $a^3-a^2-a-6=0$; $S_3$; $h_K=2$; $\varepsilon=2a^2+3a+5$; LMFDB 3.1.1099.1
1101
1:
9.183958406333525370534675429362348014648325699015021569620643581314521700665728989929492464385618565
comment: $a^3-a^2-9a+12=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2+2a-5$, $\varepsilon_2=2a^2+4a-7$; LMFDB 3.3.1101.1
-1107
1:
3.291747650180819948940358916720065510147431635966879107371281728330524709948241991158102277192748671
comment: $a^3+6a-3=0$; $S_3$; $h_K=2$; $\varepsilon=4a^2+2a+25$; LMFDB 3.1.1107.1
-1108
1:
9.474165892583798929969552936689537779657001007295861993592542815053992122814775397693721668504957287
comment: $a^3-a^2-8a+14=0$; $S_3$; $h_K=1$; $\varepsilon=490a^2-1998a+2229$; LMFDB 3.1.1108.1
1129
1:
6.727553013120447307725741309578644523865283076589315589069843861328748590035159286888821311910137552
comment: $a^3-7a-3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-2a-1$, $\varepsilon_2=a^2-8$; LMFDB 3.3.1129.1
-1135
1:
4.156682388670284069012890066954190284120197029236220073817689586802545433081080550867139194415591012
comment: $a^3-a^2-6a+11=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2-11a+12$; LMFDB 3.1.1135.1
-1144
1:
8.795578526481764063006973191948420788502796792432417855894544635665977228953462815628838862240668942
comment: $a^3-a^2+6a+2=0$; $S_3$; $h_K=1$; $\varepsilon=955a^2-1253a+6121$; LMFDB 3.1.1144.1
-1147
1:
6.603848338203641627586017569399339694276497524215922426710610449649738955213077344991404830085818124
comment: $a^3-a^2-3a+8=0$; $S_3$; $h_K=1$; $\varepsilon=49a^2-154a+183$; LMFDB 3.1.1147.1
-1164
1:
6.882377506131129641184823215752243895501354624008739299090786137516839133976604286140534234089472169
comment: $a^3-a^2-a+7=0$; $S_3$; $h_K=1$; $\varepsilon=81a^2-225a+319$; LMFDB 3.1.1164.1
-1172
1:
7.524585398382491178027430785045168506727074264998033472941071225929329014549970061710690979515644909
comment: $a^3-a^2+5a+11=0$; $S_3$; $h_K=1$; $\varepsilon=141a^2-331a+1151$; LMFDB 3.1.1172.1
-1175
1:
2.031261242232284266880141781987405288081617386471225432381649333731941834703725761041650130681278147
comment: $a^3+5a-5=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+6$; LMFDB 3.1.1175.1
-1176
1:
3.710598970947678375086536763045585083832478907614451031247539946566725345067460622991856036483987259
comment: $a^3-a^2-2a-6=0$; $S_3$; $h_K=3$; $\varepsilon=3a^2+5a+7$; LMFDB 3.1.1176.1
-1187
1:
5.424425714054846218616408350499897135598562431432076597226532567036529099707888464231284223961786439
comment: $a^3-a^2+7a-2=0$; $S_3$; $h_K=1$; $\varepsilon=34a^2-24a+231$; LMFDB 3.1.1187.1
-1188
1:
9.566826341958055508157891179045013050121430068821782408006208916516335020425244232531321638943362712
comment: $a^3+6a-12=0$; $S_3$; $h_K=1$; $\varepsilon=1144a^2+1682a+9337$; LMFDB 3.1.1188.1
-1191
1:
4.145398589127935686763518634023124259495152610117222668504121579518842626152318770918190068024125462
comment: $a^3-a^2-10a-11=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+6a+5$; LMFDB 3.1.1191.1
-1192
1:
4.989552477502075492353387071060970925745279429925979802378342048471697346668226763065821304335969721
comment: $a^3-a^2+2a+6=0$; $S_3$; $h_K=2$; $\varepsilon=16a^2-36a+77$; LMFDB 3.1.1192.1
-1196
1:
4.314454497161667833158289329782660594319550903099651932665324122273160413678723102417072975290009104
comment: $a^3-a^2+5a-7=0$; $S_3$; $h_K=1$; $\varepsilon=10a^2+3a+54$; LMFDB 3.1.1196.1
-1203
1:
8.438582135501581399081013613082075621169101883745529515440174032689016773988510509317485872181195220
comment: $a^3-a^2-3a-6=0$; $S_3$; $h_K=1$; $\varepsilon=304a^2+553a+647$; LMFDB 3.1.1203.1
-1207
1:
4.159618505394977112121005632076720638601259613594686637362773544693245281856159648169824681922949267
comment: $a^3-a^2-6a-7=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2+7a+6$; LMFDB 3.1.1207.1
-1208
1:
7.900626317218993992053978613902797396434677061363827259300831648211656992014148829708406778119868799
comment: $a^3-a^2+8a+8=0$; $S_3$; $h_K=1$; $\varepsilon=229a^2-421a+2185$; LMFDB 3.1.1208.1
-1219
1:
6.470741992155215443790283921298831123634455479737791042172938111574863031674136513255161497142590356
comment: $a^3-8a-11=0$; $S_3$; $h_K=1$; $\varepsilon=25a^2+84a+82$; LMFDB 3.1.1219.1
-1228
1:
2.255007227826675462936314646800927209933523584792531619415918328958443247030316920676849438954731049
comment: $a^3-a^2+a-7=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+a+3$; LMFDB 3.1.1228.1
-1228
2:
1.927995373115098825249608736830308153062636434347098574800845329299353291157164697117352951281943144
comment: $a^3-a^2+7a-1=0$; $S_3$; $h_K=3$; $\varepsilon=a^2-a+7$; LMFDB 3.1.1228.2
-1228
3:
2.004493982818267134718201074692934297672210626168239207011827259832393939929340903753557219623928026
comment: $a^3+4a-6=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+a+5$; LMFDB 3.1.1228.3
1229
1:
8.231918988700886290294825464139549393082314534040916325557654627221293738653100964367877566182483379
comment: $a^3-a^2-7a+6=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-1$, $\varepsilon_2=9a^2+16a-19$; LMFDB 3.3.1229.1
-1231
1:
4.633313006608119904897984694706013593652317388424306204045815565068268029835338365402022668364251058
comment: $a^3-a^2-4a+9=0$; $S_3$; $h_K=1$; $\varepsilon=6a^2-20a+23$; LMFDB 3.1.1231.1
-1235
1:
5.793106574379790640881874067245412065612277036609039928734206042743501090434093241583134234163342015
comment: $a^3-a^2-5a+10=0$; $S_3$; $h_K=1$; $\varepsilon=17a^2-60a+67$; LMFDB 3.1.1235.1
-1236
1:
11.52563430297513383956901149091940509392124349375805928774557133506084531547075147426460286086458345
comment: $a^3-a^2+4a+12=0$; $S_3$; $h_K=1$; $\varepsilon=7209a^2-18217a+56653$; LMFDB 3.1.1236.1
-1255
1:
2.145418550845346305956867360412933842671418961884364592154717039025452962274947252907912209323359627
comment: $a^3-a^2+4a+5=0$; $S_3$; $h_K=2$; $\varepsilon=a^2-2a+6$; LMFDB 3.1.1255.1
1257
1:
6.196890475052458051405396521689689771958154323139228708774529549822704244404149127001640442220408420
comment: $a^3-a^2-8a+9=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-1$, $\varepsilon_2=a^2-2a-2$; LMFDB 3.3.1257.1
-1259
1:
5.459820908682908694212713902070894732196858878123423418116269293683840396880947301261989149100130792
comment: $a^3-a^2-9a+16=0$; $S_3$; $h_K=1$; $\varepsilon=8a^2-34a+39$; LMFDB 3.1.1259.1
-1267
1:
7.610339219163543721974980114647431905613689410146268802134852244745985587098696233461840684756852491
comment: $a^3-a^2+7a-4=0$; $S_3$; $h_K=1$; $\varepsilon=294a^2-120a+1987$; LMFDB 3.1.1267.1
-1272
1:
12.16617209232756212079147881345102696116103763233674250989684027693068896423623310839437234084554960
comment: $a^3-a^2-3a+15=0$; $S_3$; $h_K=1$; $\varepsilon=9040a^2-31909a+53602$; LMFDB 3.1.1272.1
-1291
1:
7.375228296139726499572407452611678648245031280230716492983960005123735854623314322192052746933125236
comment: $a^3-2a-7=0$; $S_3$; $h_K=1$; $\varepsilon=120a^2+271a+372$; LMFDB 3.1.1291.1
-1292
1:
4.534100415356625226495675800555975043972045083258035190092115227815752980952189180581419022843835136
comment: $a^3-10a-14=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2+11a+11$; LMFDB 3.1.1292.1
-1295
1:
3.527249809544154673859251434161958652291933913717839758670993702765792236962757530374242567497310585
comment: $a^3-a^2+7=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2-8a+13$; LMFDB 3.1.1295.1
-1300
1:
10.13986302770529995555968415931289446809532763805332621381341652732983484875208208623687394598887802
comment: $a^3-a^2-3a-13=0$; $S_3$; $h_K=1$; $\varepsilon=1185a^2+2611a+4809$; LMFDB 3.1.1300.1
1300
1:
6.550005465820169148134495440204736753673778196081862401095191413605595464869649778446478547740564610
comment: $a^3-10a-10=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=a^2+2a-1$; LMFDB 3.3.1300.1
1304
1:
11.92653790077388184619408246282243659082278047967086523585168698100401133577084923096615679784037203
comment: $a^3-11a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=\tfrac{1}{2}(a^2-a-8)$, $\varepsilon_2=12a^2-2a-131$; LMFDB 3.3.1304.1
-1315
1:
7.805891497564325660914847388041307322362830732835406107391515738528168332679077869305793391740721666
comment: $a^3+7a-12=0$; $S_3$; $h_K=1$; $\varepsilon=196a^2+266a+1733$; LMFDB 3.1.1315.1
-1316
1:
7.746284026286713450988454715521643927372315789217792906749733842282656918680374530682868158063355568
comment: $a^3-a^2+9a+7=0$; $S_3$; $h_K=1$; $\varepsilon=196a^2-331a+1992$; LMFDB 3.1.1316.1
-1319
1:
2.444778580529881583009554656344767662192847188620301937769521184042886281934276789595411860230761070
comment: $a^3-a-7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2a+3$; LMFDB 3.1.1319.1
-1323
1:
2.441056470389030485120336455470934698939707877727826778518897386755290011009830218459429210390645871
comment: $a^3-7=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+2a+4$; LMFDB 3.1.1323.1; $K=\mathbb{Q}(\sqrt[3]{7})$, a pure cubic field
-1327
1:
3.695699825214975292534138468234280896232635196685251640835394888154071509140164829738135693992540443
comment: $a^3+a-7=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+7a+16$; LMFDB 3.1.1327.1
-1336
1:
9.568084973681145930825870487782299401373021077217115570503935904681966231205189794694554939143033633
comment: $a^3-a^2+9a-13=0$; $S_3$; $h_K=1$; $\varepsilon=1204a^2+443a+11442$; LMFDB 3.1.1336.1
1345
1:
4.922530816408729483591228941884431930118634664737777974535419179628032397591057476361276706205928813
comment: $a^3-7a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a^2-2a-2$; LMFDB 3.3.1345.1
-1347
1:
9.033362677905061002566166833547581590635839255266873211252813114678420141742434459100349260396593368
comment: $a^3-a^2-7a-8=0$; $S_3$; $h_K=1$; $\varepsilon=345a^2+890a+771$; LMFDB 3.1.1347.1
-1351
1:
3.534081284399757470029666085908559277107937288356638940427765430818479877925338872503574404919327698
comment: $a^3-a^2-7=0$; $S_3$; $h_K=1$; $\varepsilon=3a^2+4a+9$; LMFDB 3.1.1351.1
-1355
1:
6.413518080220634337233081242309932385056010963045288743321853791422017984228639405084208100379236409
comment: $a^3+2a-7=0$; $S_3$; $h_K=1$; $\varepsilon=65a^2+102a+290$; LMFDB 3.1.1355.1
-1356
1:
2.721405552627277936366666651072317088260690882719430608704994096362069479067350677432503090364160675
comment: $a^3-a^2+a+21=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{3}(2a^2-7a+18)$; LMFDB 3.1.1356.1
-1356
2:
3.125990921744873555378315433306309609510828559738256825955275921739503840315058963188452104048516842
comment: $a^3-a^2+3a-15=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(3a^2+4a+19)$; LMFDB 3.1.1356.3
-1356
3:
2.406785035626950884776407291014536712806540968898639396397114657614565095623150434986033557614877756
comment: $a^3-a^2+11a+1=0$; $S_3$; $h_K=3$; $\varepsilon=a^2-a+11$; LMFDB 3.1.1356.2
-1363
1:
7.215221596267840927797473497202021345896240286716041159814342966612498908829056738300258864230205420
comment: $a^3-a^2-9a-10=0$; $S_3$; $h_K=1$; $\varepsilon=46a^2+135a+117$; LMFDB 3.1.1363.1
1369
1:
3.126230190914987042161074113457852092068986307569090854416158604791185635391182158506084741951463283
comment: $a^3-a^2-12a-11=0$; $C_3$, conductor $37$; $h_K=1$; $\varepsilon_1=a^2-2a-9$, $\varepsilon_2=a+1$; LMFDB 3.3.1369.1; the cubic subfield of $\mathbb{Q}(\zeta_{37})$
-1371
1:
2.138334399090961934419449870414122969739377588556269551435662146293658485686354859344796406192653619
comment: $a^3-a^2+3a+6=0$; $S_3$; $h_K=4$; $\varepsilon=a^2-2a+5$; LMFDB 3.1.1371.1
1373
1:
9.422565053416303701487182866547319633731816393866786675769295771920796429129428798956655168666486930
comment: $a^3-8a-5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-6$, $\varepsilon_2=2a^2-2a-13$; LMFDB 3.3.1373.1
-1383
1:
5.011567618660187719055153339102543769631640989558551462772278917176205185468144418105468118028930615
comment: $a^3-a^2+6a+3=0$; $S_3$; $h_K=1$; $\varepsilon=20a^2-29a+133$; LMFDB 3.1.1383.1
1384
1:
10.38253452348559038184878591714721608000651642533432127783321685736733137587701783494613123490015609
comment: $a^3-a^2-10a+14=0$; $S_3$; $h_K=1$; $\varepsilon_1=2a-3$, $\varepsilon_2=a^2-a-5$; LMFDB 3.3.1384.1
-1388
1:
4.674311195494111218227126900471975306344822774706021728342286255410673195103893622509106769039681579
comment: $a^3-a^2+7a-5=0$; $S_3$; $h_K=1$; $\varepsilon=15a^2-4a+102$; LMFDB 3.1.1388.1
1396
1:
8.146464149133815714157885825571536490201685969138733529294828027661738049446372820645128107894008109
comment: $a^3-a^2-7a+5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-8$, $\varepsilon_2=a^2-a-9$; LMFDB 3.3.1396.1
-1399
1:
1.948804579651008279265953168185057335853631117652597726546065491275661851860403643771731121032766831
comment: $a^3+7a-1=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+7$; LMFDB 3.1.1399.1
-1407
1:
5.370659761352341223770301095900109075324465263900762499941665143583797986291287062433651546827960375
comment: $a^3-a^2-8a-9=0$; $S_3$; $h_K=1$; $\varepsilon=8a^2+22a+19$; LMFDB 3.1.1407.1
-1419
1:
8.619385964843155273946757666737168329945862035615099010382496282878011345493668608218936611050661832
comment: $a^3-a^2+10a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(939a^2+165a+9584)$; LMFDB 3.1.1419.1
-1420
1:
7.060444682833653921101731948142572513528267366234811285167124963139263282549010033015256038824995476
comment: $a^3-a^2-5a+17=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(93a^2-358a+555)$; LMFDB 3.1.1420.1
-1423
1:
1.986919830260292653568812881037904774629180813275208011573749793033146778543875717549990546713053255
comment: $a^3-a^2+6a-7=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+6$; LMFDB 3.1.1423.1
1425
1:
6.676263981196943096081161967059275199393951825671146814220805161010043356015859263172286911095119364
comment: $a^3-a^2-8a-3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+2$, $\varepsilon_2=a^2-2a-1$; LMFDB 3.3.1425.1
-1427
1:
5.429768651919875661153150775846169887109141621108331224220718029061108253869938211856793490679536742
comment: $a^3-a^2+3a-8=0$; $S_3$; $h_K=1$; $\varepsilon=25a^2+20a+111$; LMFDB 3.1.1427.1
-1431
1:
3.800694423962798733115699972784174806831140326968028892993953188083094816222371670242643246055656577
comment: $a^3+3a-7=0$; $S_3$; $h_K=1$; $\varepsilon=5a^2+7a+25$; LMFDB 3.1.1431.1
-1432
1:
11.10791533498167194676403014287873148149166989694163074832462491514413521359825221849294734343261880
comment: $a^3+10a-8=0$; $S_3$; $h_K=1$; $\varepsilon=5692a^2+4307a+60179$; LMFDB 3.1.1432.1
1436
1:
12.70004567524769288637950570427282464896790929893034896980918375171622506779497989767979758016767946
comment: $a^3-11a-12=0$; $S_3$; $h_K=1$; $\varepsilon_1=2a+5$, $\varepsilon_2=2a^2-6a-11$; LMFDB 3.3.1436.1
-1439
1:
13.21523968417324570575296510759210323628864077684255546921428297612989319834278904089999702689135221
comment: $a^3+11a-4=0$; $S_3$; $h_K=1$; $\varepsilon=48181a^2+17317a+536215$; LMFDB 3.1.1439.1
-1448
1:
9.418571866525149398278734878348817074294695511478491669911555592733823175467503221372992205111657453
comment: $a^3+5a-14=0$; $S_3$; $h_K=1$; $\varepsilon=873a^2+1521a+7015$; LMFDB 3.1.1448.1
-1452
1:
8.295791072731404525014633443862382067162488725877449846685984143531444265671475476345620452654447355
comment: $a^3-a^2-7a+13=0$; $S_3$; $h_K=1$; $\varepsilon=164a^2-643a+730$; LMFDB 3.1.1452.1; $K=\mathbb{Q}(\sqrt[3]{44})$, a pure cubic field
-1464
1:
12.83729388385359710092342598240794343699777694717756102860081059201101077424443510790863081502158783
comment: $a^3-a^2-7a+19=0$; $S_3$; $h_K=1$; $\varepsilon=12976a^2-53793a+78378$; LMFDB 3.1.1464.1
-1480
1:
11.85632363625130625676697888862081848057329499849231053445101189620674172053515141713130049323769382
comment: $a^3+7a-2=0$; $S_3$; $h_K=1$; $\varepsilon=19473a^2+5501a+137865$; LMFDB 3.1.1480.1
-1484
1:
10.38291568986130010826651394754696759043813059079779431846022983852128886062014907581148046774161624
comment: $a^3+8a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(5081a^2+6370a+48634)$; LMFDB 3.1.1484.1
1489
1:
3.360643546045871549851155077950509111676575618591805275551330566102600704592656241490570548300183527
comment: $a^3-a^2-10a-7=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=a+2$; LMFDB 3.3.1489.1
-1491
1:
9.409681651242763686080363963853009480449704332357099776965807188319100766712302012091075250013159801
comment: $a^3-a^2+6a+12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(1757a^2-4081a+15940)$; LMFDB 3.1.1491.1
1492
1:
7.646368355074671199640020598771623380956866681941724520300761028294254528837209063307006076842348900
comment: $a^3-a^2-9a-5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+2$, $\varepsilon_2=a^2-a-1$; LMFDB 3.3.1492.1
1509
1:
11.29565829479173615401357706962464576071468120152149555875181574674898423019431210007964180299322993
comment: $a^3-a^2-7a+4=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=a^2-6a+3$; LMFDB 3.3.1509.1
-1512
1:
11.77199966587119808364329300187947775909530403913804957928306960978621703579617482183427632031056560
comment: $a^3-6a-16=0$; $S_3$; $h_K=1$; $\varepsilon=4877a^2+16069a+23683$; LMFDB 3.1.1512.1
-1515
1:
10.43229070677291307187921858624244149719776813446847312098868614431565559338144501989367486023223317
comment: $a^3-a^2+5a-8=0$; $S_3$; $h_K=1$; $\varepsilon=4113a^2+1754a+23067$; LMFDB 3.1.1515.1
1524
1:
10.44956651739565894204293538379161049037464912931645040615917625245181251261942150759840201422070289
comment: $a^3-a^2-7a+1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=6a^2+11a-6$; LMFDB 3.3.1524.1
-1539
1:
9.163564333390244547803624085692332919573249818249933167727858487298825441916348518512366874799982858
comment: $a^3+6a-5=0$; $S_3$; $h_K=1$; $\varepsilon=1234a^2+938a+8117$; LMFDB 3.1.1539.1
-1547
1:
6.461580465940407631915537206267116000574614543698807405574707419045792400650175662584529664218152774
comment: $a^3-a^2-a+8=0$; $S_3$; $h_K=1$; $\varepsilon=49a^2-140a+211$; LMFDB 3.1.1547.1
1556
1:
8.375844913311121393745851950053611272786504442155491824654899348036386989082300001548981290113468450
comment: $a^3-a^2-9a+11=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-8$, $\varepsilon_2=2a^2-19$; LMFDB 3.3.1556.1
-1559
1:
14.60210755205463011033183939239794903041771312598801506310117854801880143661071959344623355216475005
comment: $a^3-a^2-2a+16=0$; $S_3$; $h_K=1$; $\varepsilon=104269a^2-360679a+678415$; LMFDB 3.1.1559.1
-1563
1:
2.048832959943410163985856273040929791216561995607991280011805223359650483011998683155731403256342226
comment: $a^3-a^2+7a-6=0$; $S_3$; $h_K=5$; $\varepsilon=a^2+7$; LMFDB 3.1.1563.1
-1567
1:
4.975590614224634520890588367163067120159464460337351947378019146558524355236159186539963337116706841
comment: $a^3-a^2-2a-7=0$; $S_3$; $h_K=1$; $\varepsilon=10a^2+17a+26$; LMFDB 3.1.1567.1
-1572
1:
2.856784709375181866543362821621221736134986861342663992602173125577671899896303215508482347787007138
comment: $a^3-a^2+a+15=0$; $S_3$; $h_K=5$; $\varepsilon=a^2-3a+7$; LMFDB 3.1.1572.1
1573
1:
8.444941302318275573757971586320810879327691338062437834048287475584989292820329923796443484977780922
comment: $a^3-a^2-7a+2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=a^2-4a+1$; LMFDB 3.3.1573.1
-1579
1:
7.381486385918955982204616090531236626382056394453593137191040404500647563762104534957629861928020257
comment: $a^3+4a-7=0$; $S_3$; $h_K=1$; $\varepsilon=184a^2+231a+1026$; LMFDB 3.1.1579.1
-1580
1:
5.263360811914394741363120965139249601261354589990904252157278071706529316306512345528549043712895959
comment: $a^3-a^2+5a+5=0$; $S_3$; $h_K=1$; $\varepsilon=23a^2-41a+147$; LMFDB 3.1.1580.1
-1583
1:
2.859994580914755753397309422872316441638051554265976774818198803124296179722836561386707526061809591
comment: $a^3-a^2-4a-7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+2a+2$; LMFDB 3.1.1583.1
-1588
1:
5.733707674427117423844336141690658118312397162263883472915856093050839029316739620034180538000959208
comment: $a^3-11a-16=0$; $S_3$; $h_K=2$; $\varepsilon=9a^2+35a+37$; LMFDB 3.1.1588.1
1593
1:
6.330794304443486093288834600346212924238939029805301713258161657811643237267392922682231543990152356
comment: $a^3-9a-7=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=2a+5$; LMFDB 3.3.1593.1
-1599
1:
4.463921964451226205739380753994623809432103673527061737554582746292252288666320439128529729565955051
comment: $a^3+3a-23=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2+10a+37$; LMFDB 3.1.1599.1
-1603
1:
8.711441094992913029489211100617053643430462919048938026119731785354011162292049366768318586024828136
comment: $a^3-5a-16=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(483a^2+1531a+2438)$; LMFDB 3.1.1603.1
-1607
1:
15.17584049175081938358093536803402363509426271583113276224422659453167704991734623391495773065334900
comment: $a^3-a^2+2a-16=0$; $S_3$; $h_K=1$; $\varepsilon=228360a^2+365162a+1405799$; LMFDB 3.1.1607.1
-1612
1:
7.126863066774562054307457817756787271820555580646241988651847138882536519970947368301960732855690530
comment: $a^3-a^2-3a-7=0$; $S_3$; $h_K=1$; $\varepsilon=77a^2+145a+187$; LMFDB 3.1.1612.1
-1615
1:
4.501903219496251481926551913863941923009084185047114302057260165931530899059585640815270397407424581
comment: $a^3+7a-3=0$; $S_3$; $h_K=1$; $\varepsilon=12a^2+5a+86$; LMFDB 3.1.1615.1
-1619
1:
5.627258663163691084636484405470149305583316302540229823000404007222190961812788579613092619429003584
comment: $a^3-a^2+a-8=0$; $S_3$; $h_K=1$; $\varepsilon=25a^2+30a+91$; LMFDB 3.1.1619.1
-1620
1:
4.139337891150512711347988767590387915204693446226574782397010381081492277682207918447410875578772264
comment: $a^3-3a-8=0$; $S_3$; $h_K=3$; $\varepsilon=4a^2+10a+13$; LMFDB 3.1.1620.1
1620
1:
10.16794283352600619477280029264620427059909501455941905712781049781560609238134291183055494856499622
comment: $a^3-12a-14=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-9$, $\varepsilon_2=a^2+5a+5$; LMFDB 3.3.1620.1
-1647
1:
4.046729737029387216465459254995760004074326021684769581005443664801058008141600662630725469290652803
comment: $a^3-9a-13=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+7a+7$; LMFDB 3.1.1647.1
-1668
1:
14.81783842576296529309158374000394135322397362517356260395346694815899401662196475658325675055972434
comment: $a^3-a^2-4a+10=0$; $S_3$; $h_K=1$; $\varepsilon=150577a^2-512095a+627173$; LMFDB 3.1.1668.1
-1675
1:
8.659212156733210292125167897138793895118805234613908372354077570744378123229968731691226717150597490
comment: $a^3-a^2+7a+2=0$; $S_3$; $h_K=1$; $\varepsilon=742a^2-944a+5451$; LMFDB 3.1.1675.1
-1687
1:
4.913896480325573287408364353983576560975851815396846365781831088514119149043895981957055927814861513
comment: $a^3-5a-9=0$; $S_3$; $h_K=1$; $\varepsilon=7a^2+20a+22$; LMFDB 3.1.1687.1
-1691
1:
12.22753078613923880159318260614672833091881050268176523668728992117543613594851793085045373617611103
comment: $a^3-a^2+a-24=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{3}(25402a^2+54109a+194769)$; LMFDB 3.1.1691.1
-1700
1:
9.116360479088734537493263136239833436371232013775348839448010764376857896870719137891722996982583103
comment: $a^3-a^2+12a-8=0$; $S_3$; $h_K=1$; $\varepsilon=757a^2-243a+8919$; LMFDB 3.1.1700.1
-1704
1:
13.60031657733942660485932614125771464349034673984377225488911831360357103815068941243351169620488118
comment: $a^3-a^2+5a-17=0$; $S_3$; $h_K=1$; $\varepsilon=52836a^2+64071a+405946$; LMFDB 3.1.1704.1
-1708
1:
7.887958914249672195896949315599775263662301791215029242241679553078653460451501273860429857038067647
comment: $a^3-a^2-5a+11=0$; $S_3$; $h_K=1$; $\varepsilon=132a^2-473a+562$; LMFDB 3.1.1708.1
1708
1:
12.87300377430546592543164609617162139457172017429076025057622673840582687737252987262967174575660685
comment: $a^3-a^2-8a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-7$, $\varepsilon_2=5a^2-9a-29$; LMFDB 3.3.1708.1
-1720
1:
13.06757555709908336498569603840603383021176037188141760825348907612203772022645819476431240503549016
comment: $a^3-2a-16=0$; $S_3$; $h_K=1$; $\varepsilon=22281a^2+62020a+128073$; LMFDB 3.1.1720.1
-1727
1:
15.42304245518764913572575628783221496031264562627001468468240314846652173361493361615018262093093277
comment: $a^3-a-16=0$; $S_3$; $h_K=1$; $\varepsilon=248289a^2+658465a+1497967$; LMFDB 3.1.1727.1
-1732
1:
5.989168653328698677591511648318801719176675477882531501199944838657194591628365405455895495741566976
comment: $a^3+a-8=0$; $S_3$; $h_K=2$; $\varepsilon=36a^2+66a+157$; LMFDB 3.1.1732.1
-1736
1:
9.006142726023569361678857185999916767341125786722841841048551205067287633599141431610729678130685163
comment: $a^3+2a-16=0$; $S_3$; $h_K=1$; $\varepsilon=472a^2+1065a+3347$; LMFDB 3.1.1736.1
-1740
1:
10.36982834449302753010897808753809519621588421064700195740648145680592283640797246334793947015210834
comment: $a^3-a^2-5a-15=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(2437a^2+6272a+10229)$; LMFDB 3.1.1740.1
-1743
1:
6.141926401562239984576760830287073049270594581997260152823618943467170346263023511471531214812497442
comment: $a^3-a^2-8a+15=0$; $S_3$; $h_K=1$; $\varepsilon=17a^2-70a+82$; LMFDB 3.1.1743.1
-1751
1:
6.654144218999556975184674479034589412512751956188512957007807090909312046920958903580197585798427940
comment: $a^3-a^2-4a-23=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{3}(77a^2+210a+475)$; LMFDB 3.1.1751.1
-1755
1:
9.016148598375674738300032529895236123331769908359226895214884170749510074262702782774924297380449056
comment: $a^3+3a-16=0$; $S_3$; $h_K=1$; $\varepsilon=497a^2+1057a+3739$; LMFDB 3.1.1755.1
-1759
1:
18.47452151326673394158428056101509084649688980198097924244641080426970508362071944230440292418735163
comment: $a^3-a^2+10a+8=0$; $S_3$; $h_K=1$; $\varepsilon=8148721a^2-13958233a+91438529$; LMFDB 3.1.1759.1
-1763
1:
6.018670179064856901242148993097924546950875758649650838304000761144989078114181925860773997141867387
comment: $a^3+11a-8=0$; $S_3$; $h_K=1$; $\varepsilon=33a^2+23a+379$; LMFDB 3.1.1763.1
1765
1:
9.444954474038391332815547833585468650512419066682117598297889843596042017367598601250155003939215476
comment: $a^3-a^2-11a+16=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=a^2-6a+7$; LMFDB 3.3.1765.1
-1772
1:
5.186792462229249611664167354933957254895362560662929417619880459848062561581781166395970352375003376
comment: $a^3-a^2+3a+7=0$; $S_3$; $h_K=1$; $\varepsilon=18a^2-40a+103$; LMFDB 3.1.1772.1
1772
1:
15.37135371837269279123632441246239119296651044744172258506440849903952965029518261592315134449528038
comment: $a^3-a^2-12a+8=0$; $S_3$; $h_K=1$; $\varepsilon_1=\tfrac{1}{2}(a^2-a-10)$, $\varepsilon_2=27a^2-9a-329$; LMFDB 3.3.1772.1
-1807
1:
4.603667805725010542193933211284114809720523744479940005113590893207322786439521339224745044775754629
comment: $a^3-a^2+8a-3=0$; $S_3$; $h_K=1$; $\varepsilon=13a^2-8a+101$; LMFDB 3.1.1807.1
-1812
1:
15.13242188245180279074782014094080112503514197466375202135227035915034773318388893752581927605845144
comment: $a^3-a^2+8a-2=0$; $S_3$; $h_K=1$; $\varepsilon=485633a^2-361263a+3792545$; LMFDB 3.1.1812.1
-1815
1:
6.732190818971749969710806371259017962765139982393008352422001473904551638287796364762310535195198211
comment: $a^3-a^2+4a-9=0$; $S_3$; $h_K=1$; $\varepsilon=89a^2+64a+466$; LMFDB 3.1.1815.1
-1823
1:
2.124961830397634073686436269826187965665018813257005775325878911692285947199203355283064776117596769
comment: $a^3+5a-7=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+a+6$; LMFDB 3.1.1823.1
1825
1:
4.488072981889130746353112441570460057115951781034394730210575956450592114124270170915311530911006511
comment: $a^3-a^2-8a+7=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-1$, $\varepsilon_2=a-3$; LMFDB 3.3.1825.1
-1836
1:
3.064432527499879059409263761475369438024154910468931271709005973443458505098925996841481998240148359
comment: $a^3-6a-10=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+3a+3$; LMFDB 3.1.1836.1
-1836
2:
2.159543321204592480348627286765153299150587396752499223422395116957257161615086533324712565442848659
comment: $a^3+6a-6=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+a+7$; LMFDB 3.1.1836.2
-1843
1:
8.933529269279030343468471691630670710511247687559901468547940582673948508823326862510843723477458597
comment: $a^3-a^2+a+8=0$; $S_3$; $h_K=1$; $\varepsilon=652a^2-1681a+3305$; LMFDB 3.1.1843.1
-1844
1:
17.52547723060683144410582179730516364421321489013136310482247141942555904994739811818736606637359634
comment: $a^3-7a-18=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(2771105a^2+9663781a+14303140)$; LMFDB 3.1.1844.1
1849
1:
18.92186438293808262342272012692600538889151220900212408477287932916776034513404154408888120617717645
comment: $a^3-a^2-14a-8=0$; $C_3$, conductor $43$; $h_K=1$; $\varepsilon_1=2a-9$, $\varepsilon_2=a^2-3a-17$; LMFDB 3.3.1849.1; the cubic subfield of $\mathbb{Q}(\zeta_{43})$
-1868
1:
4.989366828246894297056959164502094716255312920411624559394338717016818138939545006064859015045920278
comment: $a^3-a^2+7a+13=0$; $S_3$; $h_K=1$; $\varepsilon=10a^2-23a+100$; LMFDB 3.1.1868.1
-1871
1:
2.065509028529116305275974830565202068024734108977439007071372428442307971990590242294272667672501096
comment: $a^3-a^2+8a-1=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-a+8$; LMFDB 3.1.1871.1
-1879
1:
4.710907839749488552377421270579179054344182545156413197679688616318494125655858542237461036323436039
comment: $a^3-a^2-2a-16=0$; $S_3$; $h_K=4$; $\varepsilon=5a^2+11a+25$; LMFDB 3.1.1879.1
-1892
1:
10.14128313418611996749413787594388484725016197111511642351347504450001758373506342877610887823256067
comment: $a^3-a^2+a-17=0$; $S_3$; $h_K=1$; $\varepsilon=1336a^2+2411a+8098$; LMFDB 3.1.1892.1
-1895
1:
3.139334619492466265247361886790515931295047510291350234104711860223488252151588649696655021475121388
comment: $a^3-7a-11=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+3a+3$; LMFDB 3.1.1895.1
1901
1:
10.66086931182232992525721063043372437323208780310208292449855614102019610215885880415615894712522575
comment: $a^3-a^2-9a-4=0$; $S_3$; $h_K=1$; $\varepsilon_1=2a+1$, $\varepsilon_2=a^2-5$; LMFDB 3.3.1901.1
-1915
1:
10.31493497605488229445881399498344749042471193286003187475308416296013341333555311645525077877061366
comment: $a^3-a^2-6a+20=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(2121a^2-8637a+13808)$; LMFDB 3.1.1915.1
-1927
1:
2.350033819421788237458635012602935646554796422966018017311531829189817073017727738766118971791763044
comment: $a^3-a^2+2a-9=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+a+4$; LMFDB 3.1.1927.1
1929
1:
8.217600441274803952146221439207751568345467550214644170799450228497417673231527008017196114372927206
comment: $a^3-a^2-10a+13=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=5a^2+2a-47$; LMFDB 3.3.1929.1
-1931
1:
6.415196059239889419930140613031892310539114237657412756139894782820133730531994464781623984927313710
comment: $a^3-4a-9=0$; $S_3$; $h_K=2$; $\varepsilon=34a^2+92a+113$; LMFDB 3.1.1931.1
-1932
1:
10.48885437238063325121019309507478292214024316684919701468479626634332174700803737998049814724394188
comment: $a^3-a^2+7a+3=0$; $S_3$; $h_K=1$; $\varepsilon=4344a^2-6069a+32818$; LMFDB 3.1.1932.1
1937
1:
6.542242835995370956209527630734454109870381590164980671894896961966361732691458174691277375333121423
comment: $a^3-a^2-8a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a^2+a-3$; LMFDB 3.3.1937.1
1940
1:
11.08674401125157144445969087387932442538303772183962336239253093525153645620074862147735427042671718
comment: $a^3-8a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=5a^2-a-39$; LMFDB 3.3.1940.1
1944
1:
15.60051311441474442367157130201924659602996435004676953496665150211975943036241668264233986677177895
comment: $a^3-9a-6=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-3a-1$, $\varepsilon_2=2a^2-1$; LMFDB 3.3.1944.1
-1951
1:
19.24941507324691725585648757153189343496226877225599143453162562935773360190500641030540964562782906
comment: $a^3-a^2-6a-16=0$; $S_3$; $h_K=1$; $\varepsilon=8018401a^2+22002459a+34266709$; LMFDB 3.1.1951.1
-1955
1:
7.688922980718287917701100441036177950201008596170440457709034390863313806181963195242915132414971978
comment: $a^3-a^2-5a-8=0$; $S_3$; $h_K=1$; $\varepsilon=106a^2+241a+259$; LMFDB 3.1.1955.1
1957
1:
4.551450737295837538844559052517040297788479988890416171367896064966995589999068775709141296871079463
comment: $a^3-a^2-9a+10=0$; $S_3$; $h_K=2$; $\varepsilon_1=a-1$, $\varepsilon_2=a-3$; LMFDB 3.3.1957.1
-1959
1:
7.570980878624776972213370742780796762528270359465388381300150460440846098606573305054624465925702686
comment: $a^3-a^2+8a-5=0$; $S_3$; $h_K=1$; $\varepsilon=244a^2-87a+1896$; LMFDB 3.1.1959.1
-1960
1:
4.860473232477777042914038610291573000729977358400664821961788036987412059500080317555349744601764228
comment: $a^3-a^2+5a+15=0$; $S_3$; $h_K=3$; $\varepsilon=8a^2-21a+74$; LMFDB 3.1.1960.1
-1963
1:
2.261131056982392703516138138650182468003673736555192951092130277702443444053113381014070137973491704
comment: $a^3-a^2+5a+6=0$; $S_3$; $h_K=4$; $\varepsilon=a^2-2a+7$; LMFDB 3.1.1963.1
-1967
1:
8.596744323090044893903616497233945120091530652485312219371851770858471487356891974450353130254901398
comment: $a^3-a^2+2a+25=0$; $S_3$; $h_K=1$; $\varepsilon=221a^2-757a+2278$; LMFDB 3.1.1967.1
-1971
1:
11.11797003176394047400031465447048818104417703685029240324431998025941976659759125585205367806157290
comment: $a^3-9a-20=0$; $S_3$; $h_K=1$; $\varepsilon=1989a^2+7519a+10523$; LMFDB 3.1.1971.1
-1972
1:
2.582670400250819835862322691647029309098412862826414787167086821571342447272873468476496489986517662
comment: $a^3+10a-12=0$; $S_3$; $h_K=5$; $\varepsilon=a^2+a+11$; LMFDB 3.1.1972.1
-1988
1:
10.66239912704116741589997016176079243687563670192401115661683131512834058527631710728156139926885226
comment: $a^3-a^2+9a-17=0$; $S_3$; $h_K=1$; $\varepsilon=3033a^2+2054a+30742$; LMFDB 3.1.1988.1
-1999
1:
4.005017128398287281196185102338125324419729451393585903958860846917817963154411041653194486527358796
comment: $a^3-a^2+10a-16=0$; $S_3$; $h_K=5$; $\varepsilon=4a^2+2a+43$; LMFDB 3.1.1999.1
-2003
1:
6.856402146595363972242236907260832018175325172729886165561145092252935564965462423050618818899145076
comment: $a^3-a^2-3a+10=0$; $S_3$; $h_K=1$; $\varepsilon=56a^2-183a+247$; LMFDB 3.1.2003.1
2021
1:
11.51838249274660479074514254950146549725953158034554719274803869097091545605771882889208190412291353
comment: $a^3-8a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=10a^2+28a+1$; LMFDB 3.3.2021.1
-2023
1:
5.676718655744486095716907807701695588910116829140893296672871921142072341267885500722389255056923887
comment: $a^3-a^2+6a+5=0$; $S_3$; $h_K=1$; $\varepsilon=33a^2-56a+237$; LMFDB 3.1.2023.1
2024
1:
15.77098181731251880297398154745005832458122168374507776499287135769994715825590696442810574474307791
comment: $a^3-a^2-10a-6=0$; $S_3$; $h_K=1$; $\varepsilon_1=2a^2-6a-5$, $\varepsilon_2=3a^2-9a-11$; LMFDB 3.3.2024.1
-2028
1:
3.283309509031239811929159668195337793270037653084709905354346536830968586876537075308462559509925482
comment: $a^3-26=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+3a+9$; LMFDB 3.1.2028.1; $K=\mathbb{Q}(\sqrt[3]{26})$, a pure cubic field
-2036
1:
12.26156886579601990755379124211922059618967527955760144834162144917898969964525592211196903674156835
comment: $a^3-a^2+13a-5=0$; $S_3$; $h_K=1$; $\varepsilon=16677a^2-10143a+212827$; LMFDB 3.1.2036.1
-2039
1:
2.244445092297116185348669988504576098244958494574987918287269740432929965255782228036262966976933015
comment: $a^3-a^2+4a+7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+6$; LMFDB 3.1.2039.1
-2047
1:
4.318555437853612542059487213873171081307772120273916235579427608555544976356700749315365991004314510
comment: $a^3+7a-5=0$; $S_3$; $h_K=1$; $\varepsilon=9a^2+6a+67$; LMFDB 3.1.2047.1
-2051
1:
3.243024616535903315288790200252088349344649207503001725276476459790591965308534334918413416171665137
comment: $a^3-a^2-7a+14=0$; $S_3$; $h_K=2$; $\varepsilon=a^2-4a+5$; LMFDB 3.1.2051.1
-2052
1:
13.43846817354482419668841716820348679996585920596489667068319203282105059829707746651922741573175018
comment: $a^3+9a-14=0$; $S_3$; $h_K=1$; $\varepsilon=48551a^2+63471a+519935$; LMFDB 3.1.2052.1
2057
1:
6.782438913842411686946005743768969069347312174660903034966317129698158766977724382890657862041010312
comment: $a^3-11a-11=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=a^2+3a+1$; LMFDB 3.3.2057.1
-2063
1:
7.973511473928775305356063126212381589656945253307381484508132946551085471530922661985682897226807229
comment: $a^3-a^2-2a+27=0$; $S_3$; $h_K=1$; $\varepsilon=100a^2-390a+931$; LMFDB 3.1.2063.1
-2068
1:
3.361338790203251918887691847987038163596557987910829328620428345380421436880867974542721512873784538
comment: $a^3-a^2-8a-10=0$; $S_3$; $h_K=4$; $\varepsilon=a^2+3a+3$; LMFDB 3.1.2068.1
-2071
1:
19.92667596053614080424873361875427796739407980628829407613268431743548749373277041178207294458527822
comment: $a^3+7a-16=0$; $S_3$; $h_K=1$; $\varepsilon=29779641a^2+49053625a+289259607$; LMFDB 3.1.2071.1
-2075
1:
2.864902263544463601220977235642838193607182762656262352536973929892819105247631501290542692968208844
comment: $a^3-a^2-3a-8=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+2a+3$; LMFDB 3.1.2075.3
-2075
2:
2.111047982059514677320416774966093914947224474528176927893506653515245131123058199542038042799897984
comment: $a^3-a^2+7a-8=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+7$; LMFDB 3.1.2075.2
-2075
3:
4.853370815126564910419158371569208355058766271139660981455223494348266078292323519061715115293244265
comment: $a^3-10a-15=0$; $S_3$; $h_K=3$; $\varepsilon=4a^2+15a+16$; LMFDB 3.1.2075.1
2089
1:
20.76176224233844154665137188080827888115951229316767401377061599872633642857530753830556350910327889
comment: $a^3-13a-4=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-3a-1$, $\varepsilon_2=4a-15$; LMFDB 3.3.2089.1
-2091
1:
10.59418171437354616826212247071986022507296396146168520297007558823558207661909281667995145441836773
comment: $a^3-a^2-a-26=0$; $S_3$; $h_K=1$; $\varepsilon=1425a^2+3506a+10707$; LMFDB 3.1.2091.1
-2092
1:
4.858669153629283880389025868947595247924945610798862805649565315765390707339513209816757512939379815
comment: $a^3-a^2+11a-15=0$; $S_3$; $h_K=2$; $\varepsilon=\tfrac{1}{2}(19a^2+6a+217)$; LMFDB 3.1.2092.1
2101
1:
8.542654448091514689836744423033101516942333368603696970331868178462272561303873205995730033601967115
comment: $a^3-a^2-11a-8=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=2a^2-2a-15$; LMFDB 3.3.2101.1
-2104
1:
2.559444617098394830663443733249485250156747062104553968251398081594494074963200211062853302030232120
comment: $a^3-a^2+13a-1=0$; $S_3$; $h_K=5$; $\varepsilon=a^2-a+13$; LMFDB 3.1.2104.1
-2116
1:
6.398459252991931884526144071998072590589501882735590105979296490174392745760448795542242093491005943
comment: $a^3-a^2+8a-6=0$; $S_3$; $h_K=2$; $\varepsilon=73a^2-17a+571$; LMFDB 3.1.2116.1
-2132
1:
10.73005049164238930980777229073027236875926350008604066259897429662814126885801269300698938982233541
comment: $a^3-a^2-3a+19=0$; $S_3$; $h_K=1$; $\varepsilon=1876a^2-6951a+13176$; LMFDB 3.1.2132.1
-2148
1:
16.84912539011749210256826941424163187302795919836291088171733178323689199775670618319899175803372826
comment: $a^3-a^2-7a-17=0$; $S_3$; $h_K=1$; $\varepsilon=670884a^2+1949711a+2919738$; LMFDB 3.1.2148.1
-2155
1:
10.56694868271242109186307150452197395857733822860290307199100217274535052325527244610006646312031009
comment: $a^3-2a-9=0$; $S_3$; $h_K=1$; $\varepsilon=2545a^2+6104a+9550$; LMFDB 3.1.2155.1
-2159
1:
3.643779150689476267736207091093216588885856392509026812650603706156863480932604304161530548130751997
comment: $a^3-a^2-4a+11=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2-7a+9$; LMFDB 3.1.2159.1
-2167
1:
5.030779774062129069153814225088886257270692525888992525716451185692587438960006552562421638623664328
comment: $a^3-a^2-12a+23=0$; $S_3$; $h_K=1$; $\varepsilon=4a^2-19a+24$; LMFDB 3.1.2167.1
-2168
1:
9.987783151673858872695708174188500510658961532824740280909217547921217448186434622788160975554815692
comment: $a^3-a^2-3a-17=0$; $S_3$; $h_K=1$; $\varepsilon=889a^2+2114a+4474$; LMFDB 3.1.2168.1
2177
1:
7.518040240884260957499339420430634175537061358702997302626068439852308295561593810977949796013015162
comment: $a^3-a^2-8a+5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=2a^2-a-17$; LMFDB 3.3.2177.1
-2183
1:
4.029477158375600312941350377393667534926668654938385076227304192879734361148759365047573457278294281
comment: $a^3-a-9=0$; $S_3$; $h_K=2$; $\varepsilon=4a^2+9a+16$; LMFDB 3.1.2183.1
-2184
1:
16.94711377901366160792220995423646947815047260067585339314118026491611955472106939599988408738989691
comment: $a^3-a^2+13a-9=0$; $S_3$; $h_K=1$; $\varepsilon=1751856a^2-519259a+22408780$; LMFDB 3.1.2184.1
-2188
1:
2.924616718035406539551938589865261907644492499300691704918045545717957449448447352421683090152413551
comment: $a^3-a^2+3a+17=0$; $S_3$; $h_K=3$; $\varepsilon=a^2-3a+9$; LMFDB 3.1.2188.1
-2188
2:
2.729838058791174703055880735694788938141408372090858416887083520856074607001031157274398018356432205
comment: $a^3-a^2+7a-19=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+a+9$; LMFDB 3.1.2188.3
-2188
3:
2.836856388948362888238892090244283443265666649814476329291290472001430072036474624571651973268859552
comment: $a^3-8a-20=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(a^2+4a+6)$; LMFDB 3.1.2188.2
-2191
1:
2.527920060547553484939659770515633013151929123001866749461750599747477203229155722972748058727845946
comment: $a^3+a-9=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+2a+5$; LMFDB 3.1.2191.1
-2199
1:
5.389571856220362443168082908017972283415372535665078597974019333371383354565834959730174368031932939
comment: $a^3+3a-27=0$; $S_3$; $h_K=1$; $\varepsilon=9a^2+24a+91$; LMFDB 3.1.2199.1
-2200
1:
14.64237230133755894449715632356946046596797748490721642096713550233710711580923579308844978939438765
comment: $a^3-5a-10=0$; $S_3$; $h_K=1$; $\varepsilon=112477a^2+326799a+387121$; LMFDB 3.1.2200.1
-2207
1:
3.179857263675615318164385181194366965921112005391799192855530122267329011848234604845281600448763971
comment: $a^3-a^2-6a+13=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-4a+5$; LMFDB 3.1.2207.1
2213
1:
12.68082022713452093603368239737775544244960210761943928030154407715692524730741825182666374735630348
comment: $a^3-a^2-13a-12=0$; $S_3$; $h_K=1$; $\varepsilon_1=a+1$, $\varepsilon_2=4a^2+9a-1$; LMFDB 3.3.2213.1
-2219
1:
7.501660373882950307775122232372428728859602651245983001314829935915402845605218306370813985039199257
comment: $a^3+2a-9=0$; $S_3$; $h_K=2$; $\varepsilon=160a^2+282a+817$; LMFDB 3.1.2219.1
-2220
1:
12.05113082648267761777357929286499937420572425164090800779848990606924267013967996981159736234632493
comment: $a^3-a^2-11a+21=0$; $S_3$; $h_K=1$; $\varepsilon=4793a^2-22181a+27745$; LMFDB 3.1.2220.1
-2227
1:
10.16292398041899466895008609663449581247813985833392175198303778202580560318266929479868349726910608
comment: $a^3-a^2+3a+8=0$; $S_3$; $h_K=1$; $\varepsilon=2386a^2-5535a+14463$; LMFDB 3.1.2227.1
-2228
1:
10.72121693647867922467652103533879267446536828490313456113462394760596496541832813676346319412254720
comment: $a^3+5a-8=0$; $S_3$; $h_K=1$; $\varepsilon=4754a^2+5842a+30949$; LMFDB 3.1.2228.1
2228
1:
11.09416423178924451682189610825208937399554958502300176967376744619010999224326547394255300019468539
comment: $a^3-a^2-13a+9=0$; $S_3$; $h_K=1$; $\varepsilon_1=\tfrac{1}{2}(a^2+2a-5)$, $\varepsilon_2=2a^2+6a-5$; LMFDB 3.3.2228.1
2233
1:
5.522547343408442389885536205768477779279554661884297419110980977332516174292206069760393862102416729
comment: $a^3-a^2-8a+1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=a^2-3a-1$; LMFDB 3.3.2233.1
-2235
1:
13.68376385952781214297749261901544260917625619496664777963530701324951623762933993504042451768727014
comment: $a^3-a^2-5a+12=0$; $S_3$; $h_K=1$; $\varepsilon=41650a^2-151272a+189895$; LMFDB 3.1.2235.1
2241
1:
8.263506599712203288869437629066210221461531891600696066980485104813132621739592549815249493133695188
comment: $a^3-9a-5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-3a-2$, $\varepsilon_2=a^2-2a-4$; LMFDB 3.3.2241.1
-2243
1:
3.559427876065481953277753006934456554473067370066396988137616422545992396268687222841714551043708855
comment: $a^3-a^2+7a+4=0$; $S_3$; $h_K=2$; $\varepsilon=4a^2-6a+31$; LMFDB 3.1.2243.1
-2260
1:
15.47066133308749622687933080542666466284177743372894512969334294026523314192672602698306285240223839
comment: $a^3-a^2+4a-10=0$; $S_3$; $h_K=1$; $\varepsilon=508117a^2+416845a+2791281$; LMFDB 3.1.2260.1
-2264
1:
13.00823788868275505147566476174152818390325585061437025761959733137325888093930365600124868606535802
comment: $a^3-a^2-2a+10=0$; $S_3$; $h_K=1$; $\varepsilon=28009a^2-87773a+131267$; LMFDB 3.1.2264.1
-2283
1:
12.76180661396906902076344891434301696749922919849312363952893065244493180250815775334240303062595444
comment: $a^3-a^2-a+28=0$; $S_3$; $h_K=1$; $\varepsilon=12110a^2-46447a+119587$; LMFDB 3.1.2283.1
-2284
1:
5.352508982132181357714772027157446310725236091637249445386376855346618649274469851224000348650816298
comment: $a^3-a^2+11a+9=0$; $S_3$; $h_K=2$; $\varepsilon=15a^2-26a+184$; LMFDB 3.1.2284.1
-2291
1:
14.86518327128524020808641858390727741371773651660767379017684313445327790748245610176013943655079098
comment: $a^3+8a-3=0$; $S_3$; $h_K=1$; $\varepsilon=339766a^2+125283a+2764324$; LMFDB 3.1.2291.1
2292
1:
14.36227408904369655336763752207212706753350263653629732997856447245472603589157820658010317959229967
comment: $a^3-a^2-13a+1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=\tfrac{1}{2}(17a^2-72a+9)$; LMFDB 3.3.2292.1
2296
1:
14.26555094585355859121087600940268118217693609943895214073493312170756417069267203146941750656181784
comment: $a^3-a^2-14a-14=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-3a-13$, $\varepsilon_2=4a+5$; LMFDB 3.3.2296.1
2300
1:
18.11527069298695423194413273271343989483093118142752181590791254584522672469020622456019072468701519
comment: $a^3-a^2-8a+2=0$; $S_3$; $h_K=1$; $\varepsilon_1=2a+5$, $\varepsilon_2=5a^2+11a-3$; LMFDB 3.3.2300.1
-2303
1:
2.983122680359487743005680011156653708533153819490043568969401517593779912923743046385018425567699316
comment: $a^3-a^2-2a-27=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{3}(2a^2+5a+15)$; LMFDB 3.1.2303.1
-2315
1:
7.809119737710561395950520717079471186071846656090468749653844311208621139689703791042402347133482926
comment: $a^3-a^2+5a-10=0$; $S_3$; $h_K=1$; $\varepsilon=250a^2+162a+1517$; LMFDB 3.1.2315.1
-2316
1:
12.81396418080898525991579910824056376384565449975629984054545877563795451031797929842660946988934668
comment: $a^3-a^2+a+9=0$; $S_3$; $h_K=1$; $\varepsilon=29160a^2-77589a+158020$; LMFDB 3.1.2316.1
-2319
1:
4.354236303476982259207895379967301979705165701952425036281183895337313586968671709570370884581030637
comment: $a^3-a^2-12a-15=0$; $S_3$; $h_K=2$; $\varepsilon=2a^2+7a+7$; LMFDB 3.1.2319.1
-2327
1:
2.173080273515232691567927646707977486436416316019131523595404780399839896977215106302893521438368737
comment: $a^3-a^2+8a-7=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+8$; LMFDB 3.1.2327.1
-2344
1:
14.96662073222285697963307608848394542278000552791692164553531294694286127879857194431844604195051975
comment: $a^3+7a-6=0$; $S_3$; $h_K=1$; $\varepsilon=356851a^2+280985a+2719205$; LMFDB 3.1.2344.1
-2348
1:
13.81456711413908975960994183982050552806731333881926248512332785727529924100715072690190637073673924
comment: $a^3-a^2+3a+27=0$; $S_3$; $h_K=1$; $\varepsilon=39612a^2-135021a+444046$; LMFDB 3.1.2348.1
2349
1:
11.91983614375752107871813947973582210550754619633567281603444294203989851306643054766253814196575984
comment: $a^3-12a-13=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-2a-4$, $\varepsilon_2=2a^2-4a-15$; LMFDB 3.3.2349.1
-2351
1:
16.97204716942668685872936528483843166899204656744273704536345851038950264714448481705978106543067592
comment: $a^3-a^2+6a+16=0$; $S_3$; $h_K=1$; $\varepsilon=1406209a^2-3634705a+14197369$; LMFDB 3.1.2351.1
-2355
1:
13.81437991737114700618225086036323168880793405977098680722959287255784955196595325000903278695024234
comment: $a^3+12a-23=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{3}(153391a^2+243110a+2225998)$; LMFDB 3.1.2355.1
-2372
1:
10.89315900429487853241162760016179390345578442357489338871456188685428207701501655180195496052979322
comment: $a^3-a^2+8a+2=0$; $S_3$; $h_K=1$; $\varepsilon=6216a^2-7714a+51587$; LMFDB 3.1.2372.1
-2376
1:
14.90059475290069712723374711180311084880718718667752996213235492300147915147479250532531689635217828
comment: $a^3-9a-14=0$; $S_3$; $h_K=1$; $\varepsilon=99671a^2+357959a+388537$; LMFDB 3.1.2376.1
-2380
1:
10.68700103245320719603726631792884473497283357782526972696579440721259932127529273284914704499678978
comment: $a^3-a^2-a-9=0$; $S_3$; $h_K=1$; $\varepsilon=2953a^2+4885a+10013$; LMFDB 3.1.2380.1
-2387
1:
8.458296043558053649843675960226189986829499850520858449743503323971097520708227060610489885522781643
comment: $a^3-a^2+6a-20=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(561a^2+707a+4964)$; LMFDB 3.1.2387.1
-2388
1:
18.67144404594547396391751977759405056643656384202726382674954991286115040309945379765475394956762140
comment: $a^3-a^2+13a+3=0$; $S_3$; $h_K=1$; $\varepsilon=9445048a^2-11579199a+125401996$; LMFDB 3.1.2388.1
-2403
1:
10.29900366642281925723471675968940902293250505098171770144191616718495722338617668903078444622761195
comment: $a^3-6a-11=0$; $S_3$; $h_K=1$; $\varepsilon=1310a^2+4050a+4661$; LMFDB 3.1.2403.1
-2408
1:
12.11108591918501985824291271479618445249923534479725444138865936661735054183376112536489890300422352
comment: $a^3-a^2+2a-10=0$; $S_3$; $h_K=1$; $\varepsilon=15249a^2+18035a+69863$; LMFDB 3.1.2408.1
-2420
1:
12.04003574290356089939193252659792623593733848383729334958297095692235430035070413806204283712787592
comment: $a^3-a^2+4a-20=0$; $S_3$; $h_K=1$; $\varepsilon=9305a^2+14281a+73419$; LMFDB 3.1.2420.1
-2423
1:
9.172013670545227520807680621831284297481511754817547339471096944799780805137506530389746049133086215
comment: $a^3-a^2+2a-29=0$; $S_3$; $h_K=1$; $\varepsilon=365a^2+804a+3305$; LMFDB 3.1.2423.1
-2424
1:
17.41354806235784891814771313260977154315013432820407137578554640683709445614681504525291984392264797
comment: $a^3-a^2+6a+6=0$; $S_3$; $h_K=1$; $\varepsilon=3823041a^2-6900715a+28493549$; LMFDB 3.1.2424.1
2429
1:
13.27866595897970243871111813912270705751345728616111534072413209321701044385913967081816610239756704
comment: $a^3-a^2-14a-4=0$; $S_3$; $h_K=1$; $\varepsilon_1=\tfrac{1}{2}(a^2-3a-6)$, $\varepsilon_2=\tfrac{1}{2}(5a^2-19a-6)$; LMFDB 3.3.2429.1
-2440
1:
16.85154108891417156212573594539048793223382719466935001782118291969928458928769942949959218979450427
comment: $a^3+13a-6=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(3057797a^2+1389233a+40382524)$; LMFDB 3.1.2440.1
-2443
1:
5.677128936204010543623576287607336136469399074397823234710449849216935992043498907894565538142433744
comment: $a^3+4a-9=0$; $S_3$; $h_K=2$; $\varepsilon=28a^2+41a+172$; LMFDB 3.1.2443.1
-2444
1:
8.279700671478508235907610597492939917889403586287811333727601243465468123048974907747676906989000379
comment: $a^3-4a-10=0$; $S_3$; $h_K=1$; $\varepsilon=209a^2+577a+757$; LMFDB 3.1.2444.1
-2468
1:
11.46643114825538965896958864648656321074448515148104501980830314449680335737023763493930988434597071
comment: $a^3-a^2-7a+23=0$; $S_3$; $h_K=1$; $\varepsilon=3004a^2-12849a+21082$; LMFDB 3.1.2468.1
-2472
1:
19.01580342878259657662855537677668732695118956548017221083545123163615526362690665191880338852726932
comment: $a^3-a^2-8a+24=0$; $S_3$; $h_K=1$; $\varepsilon=5379879a^2-23725279a+37864225$; LMFDB 3.1.2472.1
-2476
1:
9.223748102071802734731654912532019297011195976191421782129039740014942239667741712594676760262552986
comment: $a^3-a^2-9a-19=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(567a^2+1820a+2559)$; LMFDB 3.1.2476.1
-2479
1:
4.262880983133810767357655489457179666608617814162846257963073608146247499452489872989582447931201879
comment: $a^3-11a-17=0$; $S_3$; $h_K=1$; $\varepsilon=2a^2+8a+9$; LMFDB 3.1.2479.1
-2484
1:
17.36697580537318597254941748706149907398285484117464341052042501075506585477068273289756697752013469
comment: $a^3-6a-20=0$; $S_3$; $h_K=1$; $\varepsilon=1183709a^2+4069245a+6886619$; LMFDB 3.1.2484.1
-2488
1:
3.658944484051772018356145361163942218057720157489492715740414184199427764421401070452334764753187192
comment: $a^3-a^2+6a-10=0$; $S_3$; $h_K=4$; $\varepsilon=4a^2+2a+27$; LMFDB 3.1.2488.1
-2491
1:
12.71945299775658668624381505921831068902697222016718389364355697792266775767414307571968816826828278
comment: $a^3-a^2-2a+20=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(27707a^2-100751a+210196)$; LMFDB 3.1.2491.1
-2503
1:
5.958678337824340053289868320783176981074412746811052345234292540224515446825990109593416976098530739
comment: $a^3-a^2+2a+9=0$; $S_3$; $h_K=1$; $\varepsilon=32a^2-81a+188$; LMFDB 3.1.2503.1
-2504
1:
13.56191939871511677141038195823302967356324217698296611804115942961383382346697884690146663963213199
comment: $a^3-a^2+a+19=0$; $S_3$; $h_K=1$; $\varepsilon=37113a^2-121147a+311423$; LMFDB 3.1.2504.1
2505
1:
10.68158422308386882781323205688827262692617121098371641937728583973296436580270236214789707974465851
comment: $a^3-a^2-10a-5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-3a-2$, $\varepsilon_2=2a^2-6a-7$; LMFDB 3.3.2505.1
-2508
1:
11.79950946556427053306327262742591089124985634865096302448943318587223285181490086643555933363793776
comment: $a^3-a^2-5a-9=0$; $S_3$; $h_K=1$; $\varepsilon=6209a^2+14413a+16825$; LMFDB 3.1.2508.1
-2515
1:
11.97813006650889633856898274639620035423327725661963369141710129773703613994232044007527203714695508
comment: $a^3-8a-13=0$; $S_3$; $h_K=1$; $\varepsilon=5820a^2+19981a+22038$; LMFDB 3.1.2515.1
-2516
1:
21.26431395508875070338650147622126396806814561022764771169080836516257659524458052272759755118123959
comment: $a^3-7a-12=0$; $S_3$; $h_K=1$; $\varepsilon=68659060a^2+224307406a+252193181$; LMFDB 3.1.2516.1
-2540
1:
6.434559404988791143316270671888645322988277799012881550754149830336533800280131098053396981422490730
comment: $a^3-a^2-a-19=0$; $S_3$; $h_K=1$; $\varepsilon=27a^2+59a+161$; LMFDB 3.1.2540.1
-2548
1:
5.230335762035694574669712996076235935515243914618643186757626995672539261212362541360510370405131163
comment: $a^3-a^2+12a+8=0$; $S_3$; $h_K=3$; $\varepsilon=13a^2-21a+169$; LMFDB 3.1.2548.1
-2555
1:
9.764110866438077933487988601427751706938386102456539321342860436038149561331916024202921085065882186
comment: $a^3-a^2+14a-4=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(2545a^2-1807a+35106)$; LMFDB 3.1.2555.1
2557
1:
10.72296886122720434315941004251399524695734195429061793091511294936194967480395183137541198729619832
comment: $a^3-a^2-9a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2+2a-1$, $\varepsilon_2=a^2-4a-1$; LMFDB 3.3.2557.1
-2563
1:
5.659494398327586581291721190929574078761543495219800183007555798932652428753753004982490988445099109
comment: $a^3-a^2+a-10=0$; $S_3$; $h_K=2$; $\varepsilon=22a^2+30a+93$; LMFDB 3.1.2563.1
-2575
1:
24.61374592274093234577542271128986128090438276278780797912534794067324003650770149961741777222961821
comment: $a^3-5a-20=0$; $S_3$; $h_K=1$; $\varepsilon=1743223617a^2+5787852181a+10500712471$; LMFDB 3.1.2575.1
-2579
1:
9.582316942085877009504348936148396043858170757119275465771971614932700583201964323764375546128324971
comment: $a^3-a^2-7a-10=0$; $S_3$; $h_K=1$; $\varepsilon=561a^2+1492a+1533$; LMFDB 3.1.2579.1
2589
1:
16.29170677873844581452164256621336996246357306784585752840579348111258229547460563651938919586254094
comment: $a^3-a^2-14a+12=0$; $S_3$; $h_K=1$; $\varepsilon_1=\tfrac{1}{2}(a^2+a-10)$, $\varepsilon_2=10a^2+28a-31$; LMFDB 3.3.2589.1
-2591
1:
8.781404619546151579675689852714622117546081403361618409114014875205898331199603658089501607935562618
comment: $a^3-a^2-4a+31=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{3}(581a^2-2454a+5587)$; LMFDB 3.1.2591.1
-2595
1:
2.890543152204256806590731675056309075535855640745472858655567991045495527028022669267368286865287406
comment: $a^3-a^2-10a-20=0$; $S_3$; $h_K=5$; $\varepsilon=\tfrac{1}{2}(a^2+3a+4)$; LMFDB 3.1.2595.1
2597
1:
4.795990555363511321200595341649628490302599909660670551403631801516939059583456815608235247508804561
comment: $a^3-a^2-9a+8=0$; $S_3$; $h_K=3$; $\varepsilon_1=a-1$, $\varepsilon_2=a-3$; LMFDB 3.3.2597.1
-2599
1:
4.772190016193791006014045870960562680842528647566433233506749115188410407726945117763468751788425010
comment: $a^3-a^2-4a-9=0$; $S_3$; $h_K=1$; $\varepsilon=6a^2+13a+17$; LMFDB 3.1.2599.1
-2604
1:
11.68385740542357626250473217932857763479369046665106925034365799558388730633705771140476477075369933
comment: $a^3-a^2+9a-3=0$; $S_3$; $h_K=1$; $\varepsilon=13689a^2-9009a+120121$; LMFDB 3.1.2604.1
-2619
1:
4.708311273178500770507008745060404827812935476413155907165106511907102952217134188398495245681350375
comment: $a^3+6a-29=0$; $S_3$; $h_K=2$; $\varepsilon=\tfrac{1}{3}(14a^2+34a+167)$; LMFDB 3.1.2619.1
-2627
1:
8.222544841180844596214091173778965068865708220204384521711557987247870193050210186681884857401955138
comment: $a^3-a^2+9a-2=0$; $S_3$; $h_K=1$; $\varepsilon=428a^2-331a+3777$; LMFDB 3.1.2627.1
-2631
1:
27.94701992229782072378858352958268955570002925695137980039883203672184312242614306261927977953845531
comment: $a^3-a^2+14a-8=0$; $S_3$; $h_K=1$; $\varepsilon=99023962436a^2-41437864194a+1362237822791$; LMFDB 3.1.2631.1
-2635
1:
11.46832556011162264458720767713299184459296032885437167713432605162389876294957622524121883561939108
comment: $a^3-a^2+9a-4=0$; $S_3$; $h_K=1$; $\varepsilon=10977a^2-5960a+96069$; LMFDB 3.1.2635.1
-2636
1:
8.370782979154324719608529991978430054466834553235338279345100009967351218794674072996717478943287009
comment: $a^3-4a-20=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(323a^2+1034a+2018)$; LMFDB 3.1.2636.1
2636
1:
18.37928722862072500861749508109883024382831834886358708128905322401871225521838979015781912354906266
comment: $a^3-14a-4=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-15$, $\varepsilon_2=2a^2+3a-15$; LMFDB 3.3.2636.1
-2644
1:
16.21154200405348522235522266024451142891505288884672467942929196355282906695530556161021986983069951
comment: $a^3-a^2+5a-21=0$; $S_3$; $h_K=1$; $\varepsilon=605265a^2+879183a+5182573$; LMFDB 3.1.2644.1
-2647
1:
6.077658036539942479647358023915892879808544671805480165459082194740835272425256210561141146335175033
comment: $a^3-a^2+8a+3=0$; $S_3$; $h_K=1$; $\varepsilon=48a^2-65a+407$; LMFDB 3.1.2647.1
-2660
1:
14.34685325880478468262575803953382691028039456201128203519102444584892464678965351323372639297427523
comment: $a^3-a^2+10=0$; $S_3$; $h_K=1$; $\varepsilon=119828a^2-343602a+641663$; LMFDB 3.1.2660.1
-2668
1:
9.744433240480174574195639560155045539502521196575824033697812532628546772776359738860867941219600081
comment: $a^3-2a-10=0$; $S_3$; $h_K=1$; $\varepsilon=1054a^2+2595a+4281$; LMFDB 3.1.2668.1
2673
1:
7.760102707451648705355824840094476437591023595795074132389076165731816145877387454355444558011410371
comment: $a^3-9a-3=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-8$, $\varepsilon_2=a^2-10$; LMFDB 3.3.2673.1
-2676
1:
18.45258796126624617367807770366094261070653808736213995781244881982144416531319459915640463696790292
comment: $a^3-a^2-11a-21=0$; $S_3$; $h_K=1$; $\varepsilon=2547960a^2+8894346a+11914915$; LMFDB 3.1.2676.1
2677
1:
11.15852395349973918743108616951622089804709992934359234139621335832077084663345555993618393531064695
comment: $a^3-10a-7=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-2a-2$, $\varepsilon_2=a^2-12$; LMFDB 3.3.2677.1
-2680
1:
16.90989265575231852432637029914448414175641859502182217170241120737979531701270190782069882809737401
comment: $a^3-a^2+20=0$; $S_3$; $h_K=1$; $\varepsilon=985961a^2-3370715a+8152783$; LMFDB 3.1.2680.1
-2687
1:
4.531210403243956251749725904797218423620875907630417385281991917934273080915444583523797258875076065
comment: $a^3+5a-9=0$; $S_3$; $h_K=2$; $\varepsilon=9a^2+12a+61$; LMFDB 3.1.2687.1
-2692
1:
3.204222324218554536204693799730243619174369654281422645449620343933560827760859659245936377177500683
comment: $a^3-2a-20=0$; $S_3$; $h_K=5$; $\varepsilon=a^2+3a+7$; LMFDB 3.1.2692.1
-2695
1:
2.272269284594120900015587582986429667146225790369078684609729427748756338730393291636450947743520080
comment: $a^3+7a-7=0$; $S_3$; $h_K=3$; $\varepsilon=a^2+a+8$; LMFDB 3.1.2695.1
-2696
1:
3.810112316096228818439385424821972134572816376356119434802144553301853487990663300476909857833314520
comment: $a^3-a-10=0$; $S_3$; $h_K=4$; $\varepsilon=3a^2+7a+13$; LMFDB 3.1.2696.1
-2699
1:
10.53270953128849831851131367511643859615363046389560545400227400981001046102426699392086938890032319
comment: $a^3-a-20=0$; $S_3$; $h_K=1$; $\varepsilon=1621a^2+4599a+11427$; LMFDB 3.1.2699.1
-2700
1:
3.493769421097499623588540803616742555237759247874723525547811011149189749309649405377207193524366155
comment: $a^3-20=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(3a^2+8a+22)$; LMFDB 3.1.2700.1; $K=\mathbb{Q}(\sqrt[3]{20})$, a pure cubic field
2700
1:
20.36935518703493762142763392078982764757196785139695832593683932273228452991353738498578756340979450
comment: $a^3-15a-20=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2+a-1$, $\varepsilon_2=8a^2-22a-59$; LMFDB 3.3.2700.1
-2708
1:
11.17008291483927558755620456947481673155405281971622416534169336931652383611372013262768350943268574
comment: $a^3+2a-20=0$; $S_3$; $h_K=1$; $\varepsilon=3497a^2+8636a+28321$; LMFDB 3.1.2708.1
2708
1:
12.95045095850632828031637134579280604687513694042501262943980969090505700930649766002935064918855644
comment: $a^3-a^2-11a-7=0$; $S_3$; $h_K=1$; $\varepsilon_1=3a^2-5a-29$, $\varepsilon_2=2a^2-6a-9$; LMFDB 3.3.2708.1
2713
1:
12.33564723720417037684052234004789746247882244039371529820499126080680279206937347967611935646796836
comment: $a^3-13a-15=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-2a-13$, $\varepsilon_2=2a^2-4a-17$; LMFDB 3.3.2713.1
-2723
1:
8.551013767919826678171997153763745501051167448890404586811119712204425008190156752880236645527647597
comment: $a^3+8a-5=0$; $S_3$; $h_K=1$; $\varepsilon=570a^2+341a+4764$; LMFDB 3.1.2723.1
-2727
1:
25.74768049621404318695593842709197872697290678672043815328348462451956256010630489696392073667548644
comment: $a^3+3a-20=0$; $S_3$; $h_K=1$; $\varepsilon=7780120964a^2+18272195354a+66253978837$; LMFDB 3.1.2727.1
-2728
1:
15.86334106922596274238967038986878585006698000259437130447323115267342157264622273354232359813342410
comment: $a^3+10a-16=0$; $S_3$; $h_K=1$; $\varepsilon=500448a^2+676886a+5920009$; LMFDB 3.1.2728.1
-2732
1:
6.333326953623260737089740732091942843822253108727227264592127754998052483249601848022638037686499664
comment: $a^3+2a-10=0$; $S_3$; $h_K=1$; $\varepsilon=46a^2+85a+249$; LMFDB 3.1.2732.1
-2740
1:
17.25096504031261414784369671238315644484553284030830638813559214744334262410774855095785469163352096
comment: $a^3-a^2-10=0$; $S_3$; $h_K=1$; $\varepsilon=2165776a^2+3345066a+8511559$; LMFDB 3.1.2740.1
-2747
1:
17.13848957239402342504729296064293397076012499368463551348674828089520725272471851767984097305073905
comment: $a^3-a^2+a+30=0$; $S_3$; $h_K=1$; $\varepsilon=974113a^2-3615444a+10777477$; LMFDB 3.1.2747.1
-2759
1:
2.448892259654496650539697656921417734210834344250606445633772612096986638471529980849691147669766612
comment: $a^3-a^2+4a-11=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+a+6$; LMFDB 3.1.2759.1
-2764
1:
5.801926673796759427841687233862993670446828435048886179351922503107713789916882523648024534622585218
comment: $a^3+4a-20=0$; $S_3$; $h_K=2$; $\varepsilon=\tfrac{1}{2}(35a^2+78a+314)$; LMFDB 3.1.2764.1
-2767
1:
3.072007608013725846986434000790117616850317092973150686574350932729112635716213951805095776523608584
comment: $a^3-5a-11=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+3a+4$; LMFDB 3.1.2767.1
2777
1:
3.949038016309490466058004299913238592254895267125507981009394393497230207669087144857209709009364022
comment: $a^3-a^2-14a+23=0$; $S_3$; $h_K=2$; $\varepsilon_1=a^2+a-12$, $\varepsilon_2=a^2+2a-8$; LMFDB 3.3.2777.1
-2787
1:
7.273817095399400648695463525299339944832859603898479776422938236011218896773967231281636250624589550
comment: $a^3-6a-31=0$; $S_3$; $h_K=2$; $\varepsilon=\tfrac{1}{3}(118a^2+445a+970)$; LMFDB 3.1.2787.1
-2791
1:
4.433219106213560702756632662513832981780159201586869530828705002591083587573040121691108820521294445
comment: $a^3-a^2-2a+11=0$; $S_3$; $h_K=1$; $\varepsilon=5a^2-16a+25$; LMFDB 3.1.2791.1
-2795
1:
9.645882960430725366792355633838570476063697380585476746920468237002228622985692100951659618150301618
comment: $a^3-a^2+10a-20=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(1957a^2+1493a+22202)$; LMFDB 3.1.2795.1
-2796
1:
10.55009350885661376855095282795470909567136148884287965055939123233310875847014887089488443875420043
comment: $a^3-a^2+5a-11=0$; $S_3$; $h_K=1$; $\varepsilon=3585a^2+2673a+22591$; LMFDB 3.1.2796.1
-2803
1:
10.90841157402486177518822097357083376694062486930135706600397367940114357043462334298154968120912759
comment: $a^3-a^2+7a-10=0$; $S_3$; $h_K=1$; $\varepsilon=5625a^2+1918a+41947$; LMFDB 3.1.2803.1
2804
1:
15.24005106091141614818636197910958626432023616432170451101792388241538656459074594742989606162814083
comment: $a^3-a^2-9a-1=0$; $S_3$; $h_K=1$; $\varepsilon_1=a$, $\varepsilon_2=21a^2+56a+10$; LMFDB 3.3.2804.1
-2808
1:
16.45334664212478859419801538781652502708049552850082129027337142917081421851000295966461427139052654
comment: $a^3+3a-10=0$; $S_3$; $h_K=1$; $\varepsilon=1199361a^2+2037577a+7059693$; LMFDB 3.1.2808.1
2808
1:
20.31285608631372919583361250062281624854693600122497859330148175875317675241033796292172568154890028
comment: $a^3-9a-2=0$; $S_3$; $h_K=1$; $\varepsilon_1=3a^2-9a-1$, $\varepsilon_2=2a^2-4a-1$; LMFDB 3.3.2808.1
-2819
1:
3.246106144575685971566639848256083795718100050721021675298159439802722955729800598470408090849287104
comment: $a^3-a^2+5a-32=0$; $S_3$; $h_K=5$; $\varepsilon=a^2+2a+11$; LMFDB 3.1.2819.1
-2820
1:
21.58006098767279160954030175926320834786694284900197523311092471258531889641550290532798236674910563
comment: $a^3-a^2+9a+15=0$; $S_3$; $h_K=1$; $\varepsilon=144293180a^2-326740014a+1711774439$; LMFDB 3.1.2820.1
-2824
1:
8.244067333289613041008362685646486727692435636387582357787435579851977223450280285398629365128284067
comment: $a^3-a^2-6a+14=0$; $S_3$; $h_K=2$; $\varepsilon=159a^2-611a+783$; LMFDB 3.1.2824.1
-2835
1:
4.316956728453106995185006680531247465596077654319056014269473043006068785683482027319283203618819642
comment: $a^3-12a-19=0$; $S_3$; $h_K=3$; $\varepsilon=2a^2+8a+9$; LMFDB 3.1.2835.1
2836
1:
9.691729941381213562456102744468732501926547041469513598460085425059878600062460935210970766351795209
comment: $a^3-a^2-9a+7=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-10$, $\varepsilon_2=2a^2-17$; LMFDB 3.3.2836.1
-2843
1:
3.992402158918268276394450243536647752048970322096765560311014884466702421613770286724027862316507839
comment: $a^3-a^2+a+10=0$; $S_3$; $h_K=2$; $\varepsilon=4a^2-11a+23$; LMFDB 3.1.2843.1
-2852
1:
11.42896753364027425732106275260384989303112133476952306944355138228228141691571766102701582686563829
comment: $a^3+14a-4=0$; $S_3$; $h_K=1$; $\varepsilon=6456a^2+1834a+90905$; LMFDB 3.1.2852.1
-2855
1:
3.489107740279280754457757833144018449231671633587991172873157254349674303387844143017682992287001079
comment: $a^3-a^2-10a-13=0$; $S_3$; $h_K=1$; $\varepsilon=a^2+3a+3$; LMFDB 3.1.2855.1
-2856
1:
2.815663714050794635922569981784556320530136056762886162496220143413955528912728686667067892687010709
comment: $a^3-a^2+9a-21=0$; $S_3$; $h_K=7$; $\varepsilon=a^2+a+11$; LMFDB 3.1.2856.1
2857
1:
4.869562975287536899057212818690822508577939916894004985795064488105529930681695465215220990634105444
comment: $a^3-a^2-10a+11=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-1$, $\varepsilon_2=a-3$; LMFDB 3.3.2857.1
-2859
1:
6.295300046912174801150346939868063759322981050696464667275925095471031756827192795182638501985355122
comment: $a^3-a^2+9a-6=0$; $S_3$; $h_K=2$; $\varepsilon=60a^2-19a+527$; LMFDB 3.1.2859.1
-2860
1:
11.59655859192783642305740015604968913542328311415257324335881479566921200724371955385165622186724456
comment: $a^3-a^2-a+21=0$; $S_3$; $h_K=1$; $\varepsilon=4540a^2-16206a+37103$; LMFDB 3.1.2860.1
-2867
1:
8.918111573771351921588038621104004634811116438219279198914612636570514505889756929163600667795057728
comment: $a^3-a^2+2a+20=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(711a^2-2279a+6448)$; LMFDB 3.1.2867.1
-2872
1:
16.86393001151643347483429917276497220187179140383016408844877184604447448365143989043033985635742785
comment: $a^3+13a-10=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(2881007a^2+2126981a+39023392)$; LMFDB 3.1.2872.1
-2879
1:
2.363929033812144461251570813273372968734369363052830293005632571826842433404415195359243421449083508
comment: $a^3-a^2+6a+7=0$; $S_3$; $h_K=1$; $\varepsilon=a^2-2a+8$; LMFDB 3.1.2879.1
-2888
1:
4.078975569036387011889047503357799729739830675431005523611939727889946749446830556840510957091869668
comment: $a^3-a^2+13a+7=0$; $S_3$; $h_K=3$; $\varepsilon=4a^2-6a+55$; LMFDB 3.1.2888.1
-2891
1:
5.740051320641186098611652677820075482299856858108313759417193377990367186307288660347560943414876706
comment: $a^3-a^2-9a+36=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{3}(22a^2-106a+207)$; LMFDB 3.1.2891.2
-2891
2:
3.630606628240654021340584957420198002657001103646579002949498160662613594988904296124738575631245862
comment: $a^3-a^2-2a-20=0$; $S_3$; $h_K=3$; $\varepsilon=\tfrac{1}{2}(3a^2+7a+18)$; LMFDB 3.1.2891.3
-2891
3:
2.341237389561313311104966909038383802638748415613589280671705611005877516561123105705636824027087699
comment: $a^3-a^2+5a+8=0$; $S_3$; $h_K=3$; $\varepsilon=a^2-2a+7$; LMFDB 3.1.2891.1
-2892
1:
12.52344580433220552597572378623966265826474013269574702300148716398765106691941336963608829908571106
comment: $a^3-a^2-13a-17=0$; $S_3$; $h_K=1$; $\varepsilon=6593a^2+23834a+24286$; LMFDB 3.1.2892.1
-2895
1:
8.237222125817426904886683016766721963379356860012272642600834537987950538304771285638292556479574182
comment: $a^3+3a-31=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{3}(421a^2+1189a+4621)$; LMFDB 3.1.2895.1
-2911
1:
2.222292490281387668857033072090473644465097377714473971305721295121394976430029632031163589141349016
comment: $a^3-a^2+8a-9=0$; $S_3$; $h_K=2$; $\varepsilon=a^2+8$; LMFDB 3.1.2911.1
-2915
1:
9.128478120492535114360107316323440575513728955174769423472787690463647940387334256079630440387113605
comment: $a^3-a^2-a-10=0$; $S_3$; $h_K=1$; $\varepsilon=585a^2+1006a+2151$; LMFDB 3.1.2915.1
2917
1:
11.93327521261013620557737373187519699116845607467762753414730800952812105537609331711108306651669239
comment: $a^3-a^2-13a+20=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2+2a-7$, $\varepsilon_2=a^2+6a-13$; LMFDB 3.3.2917.1
-2920
1:
18.95381177221729288644167955141881635747964588806405814840186953838324309517663991671223060184136155
comment: $a^3-a^2-6a-10=0$; $S_3$; $h_K=1$; $\varepsilon=7086655a^2+17827833a+20157219$; LMFDB 3.1.2920.1
2920
1:
17.94158608408583537632978052260253627588687137254920529937992440097563567285931184348300854035483352
comment: $a^3-a^2-15a-5=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-3a-7$, $\varepsilon_2=2a^2-8a-3$; LMFDB 3.3.2920.1
-2923
1:
11.58716060103559274959361407445125459119631081165586918778046444180666905923028203547128023628017731
comment: $a^3-a^2+14a-12=0$; $S_3$; $h_K=1$; $\varepsilon=\tfrac{1}{2}(14843a^2-2013a+206062)$; LMFDB 3.1.2923.1
2941
1:
13.71764692824372696643250081501284162815540481946647151116124993704695064116105290544482005090774723
comment: $a^3-a^2-14a+4=0$; $S_3$; $h_K=1$; $\varepsilon_1=a^2-a-13$, $\varepsilon_2=\tfrac{1}{2}(a^2-3a-22)$; LMFDB 3.3.2941.1
-2951
1:
5.707154448111095468673955301361363052333253068813722304700119548010156770984467107607638077562844834
comment: $a^3-a^2+6a-11=0$; $S_3$; $h_K=1$; $\varepsilon=29a^2+17a+201$; LMFDB 3.1.2951.1
-2955
1:
16.19998231897462619267634006702917436001045664806557508742719783469431776830224130561722169644544681
comment: $a^3-a^2-5a-30=0$; $S_3$; $h_K=1$; $\varepsilon=299446a^2+915151a+2214757$; LMFDB 3.1.2955.1
-2956
1:
12.47703444658797066553111916148991597674214686903385201830795015346525282067386285942278566182313720
comment: $a^3+4a-10=0$; $S_3$; $h_K=1$; $\varepsilon=23268a^2+36223a+149463$; LMFDB 3.1.2956.1
-2964
1:
18.79424877972720112928360624545070819124537274796243579530042436377355154595621202084693918008112638
comment: $a^3-a^2+5a+19=0$; $S_3$; $h_K=1$; $\varepsilon=7661145a^2-21832597a+78691281$; LMFDB 3.1.2964.1
2981
1:
14.62862220155472686162828217916756868820028889297425894645514776506419318443367124714962011612132719
comment: $a^3-a^2-11a+14=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=a^2+10a-15$; LMFDB 3.3.2981.1
-2991
1:
31.17759997207555333078350387448124421669891714961223681400623191521913050049048517059118935284902047
comment: $a^3-a^2-6a+24=0$; $S_3$; $h_K=1$; $\varepsilon=1110198806284a^2-4669670199610a+8310501779239$; LMFDB 3.1.2991.1
2993
1:
7.514079841906862610095209385536107704536309161082370028036274333325814308688329208514615155489653191
comment: $a^3-a^2-12a+17=0$; $S_3$; $h_K=1$; $\varepsilon_1=a-3$, $\varepsilon_2=2a-3$; LMFDB 3.3.2993.1
Definition
Let $K$ be a cubic field [2] of discriminant $D$, $\varepsilon_1,\ldots,\varepsilon_r$ fundamental units of $K$ ($r=1$ for $D<0$, $2$ for $D>0$) and $\sigma_1,\ldots,\sigma_r$ distinct real embeddings. Listed is the regulator [3] $R_K=|\det(\log|\sigma_i(\varepsilon_j)|)_{i,j\leq r}|$.
Parameters
$D$
—   discriminant of $K$ ($D$ the discriminant of a cubic field)
$k$
—   index of $K$ among the cubic fields of discriminant $D$ ($k\geq 1$)
Formulas
(1)
$\kappa_K=\frac{2\pi\,h_KR_K}{\sqrt{|D|}}$ for $D<0$ and $\kappa_K=\frac{4\,h_KR_K}{\sqrt{D}}$ for $D>0$, where $\kappa_K$ is the residue at $s=1$ of the Dedekind zeta function of $K$ and $h_K$ its class number: the class number formula [4] $\kappa_K=\frac{2^{r_1}(2\pi)^{r_2}h_KR_K}{w_K\sqrt{|D|}}$ with $w_K=2$, since a cubic field contains no root of unity other than $\pm1$. The quadratic case is the table of residues of quadratic fields.
(2)
$R_K=\frac{\sqrt{D}\,|L(1,\chi)|^2}{4h_K}$ for a cyclic cubic field, where $\chi$ is either cubic Dirichlet character of conductor $\sqrt{D}$: $\zeta_K(s)=\zeta(s)L(s,\chi)L(s,\bar\chi)$ gives $\kappa_K=L(1,\chi)L(1,\bar\chi)=|L(1,\chi)|^2$, and formula (1) does the rest. For $D=49$ and $\chi=\chi_7(2,\cdot)$ in Conrey's notation, $|L(1,\chi)|^2=0.3002598183\ldots$ and $R_K=\frac{7}{4}|L(1,\chi)|^2$.
(3)
$R_K=\log\sigma(\varepsilon)=-2\log|\tau(\varepsilon)|$ for $D<0$, where $\sigma$ is the real embedding, $\tau$ a complex one and $\varepsilon$ the fundamental unit with $\sigma(\varepsilon)>1$, because $\sigma(\varepsilon)\,|\tau(\varepsilon)|^2=|N(\varepsilon)|=1$. For $D>0$ every unit satisfies $\log|\sigma_1(\varepsilon)|+\log|\sigma_2(\varepsilon)|+\log|\sigma_3(\varepsilon)|=0$, which is why any two of the embeddings give the same determinant.
Comments
(4)
For a number field with $r_1$ real and $r_2$ complex places the regulator is the absolute value of any $r\times r$ minor of the $r\times(r+1)$ matrix $(N_i\log|\sigma_i(\varepsilon_j)|)$, where $r=r_1+r_2-1$ is the unit rank and $N_i=1$ at a real place and $2$ at a complex one; every unit has norm $\pm1$, so the rows sum to zero and the minor depends neither on the column left out nor on the choice of fundamental units. A complex cubic field has $r_1=r_2=1$: leaving out the complex place gives $R_K=|\log|\sigma(\varepsilon)||$ for the real embedding $\sigma$, and leaving out the real place gives $2|\log|\tau(\varepsilon)||$ for a complex embedding $\tau$, the same number because $|\sigma(\varepsilon)|\,|\tau(\varepsilon)|^2=1$. A totally real cubic field has $r_1=3$, and any two of its three embeddings give the determinant listed. There is no other factor: this is the convention of the LMFDB [7] and of PARI's bnfinit, and the one under which the class number formula (1) holds as written.
(5)
Non-isomorphic cubic fields can share a discriminant, which quadratic fields never do. The fields of discriminant $D$ are numbered $k=1,2,\ldots$ in lexicographic order of the coefficient vectors $(a_2,a_1,a_0)$ of their reduced defining polynomials $x^3+a_2x^2+a_1x+a_0$, the polynomial PARI's polredabs returns for the field and the LMFDB displays as its defining polynomial. Seven discriminants with $|D|\leq 3000$ carry more than one field: $D=-972$ and $-1836$ two each, and $D=-1228$, $-1356$, $-2075$, $-2188$ and $-2891$ three each; so for $D=-1228$, $k=1$ is $x^3-x^2+x-7$, $k=2$ is $x^3-x^2+7x-1$ and $k=3$ is $x^3+4x-6$. Every other negative $D$ here, and every positive one, carries a single field, with $k=1$. The LMFDB's own index for fields sharing a discriminant is not this order (for $D=-2891$ its first field is $k=3$ here), and its label knowl states no rule for it; the label in each entry's comment was read from that field's LMFDB page.
(6)
The comment on each entry gives the reduced polynomial $f$, so that $K=\mathbb{Q}(a)$ with $f(a)=0$; the Galois group of the Galois closure, $C_3$ or $S_3$; the class number $h_K$; the fundamental unit or pair, as polynomials in $a$; and the LMFDB label. For $D<0$ the unit $\varepsilon$ shown is the one of $\pm\varepsilon^{\pm1}$ whose real embedding exceeds $1$, so that $R_K=\log\sigma(\varepsilon)$ with no absolute value; its coefficients run to thirteen digits at $D=-2991$. For $D>0$ no fundamental pair is canonical, and the pair shown is the one PARI's bnfinit returned, each unit with the sign that makes its leading coefficient positive; Wikipedia and the LMFDB give other pairs for $D=49$, and they generate the same group. Cohen's tables B.3 and B.4 [1] list these fields with their class numbers and units.
(7)
$D$ is a perfect square exactly for the cyclic cubic fields, those with Galois group $C_3$, and $\sqrt{D}$ is then the conductor: the seven cyclic fields here have conductors $7$, $9$, $13$, $19$, $31$, $37$ and $43$, and each is the cubic subfield of the cyclotomic field of that conductor, for $7$ and $9$ its maximal real subfield. For a cyclic field $\zeta_K(s)=\zeta(s)L(s,\chi)L(s,\bar\chi)$ with $\chi$ a cubic Dirichlet character of that conductor, so $R_K$ is a rational multiple of $|L(1,\chi)|^2$, formula (2); $L(1,\chi)$ is held in the table of Dirichlet $L$-values for conductors up to $29$. The other 508 fields have Galois group $S_3$, and their Galois closure contains the quadratic field of discriminant $d$, where $D=df^2$ with $d$ a fundamental discriminant [2]; the pure cubic fields $\mathbb{Q}(\sqrt[3]{m})$ are those with $d=-3$.
(8)
$h_K=1$ for 423 of the 515 fields, $h_K=2$ for 41, $h_K=3$ for 36, $h_K=4$ for 6, $h_K=5$ for 8, and $h_K=7$ for one, the field of $x^3-x^2+9x-21$ with $D=-2856$. The three fields of $D=-1228$ all have $h_K=3$.
(9)
A few entries are constants a reader may hold under another name. For $D=-23$, the cubic field of smallest $|D|$, $\varepsilon$ is the plastic number $\rho=1.3247\ldots$, the real root of $x^3-x-1$ (OEIS A060006 [11]), and $R_K=\log\rho$ is OEIS A202539 [10], listed there as the solution of $e^{2x}-e^{-x}=1$, which is $\rho^3=\rho+1$ in disguise. For $D=49$, $K=\mathbb{Q}(\zeta_7)^+$ and $R_K=0.5254\ldots$ is the worked example of [3], which writes the field with the polynomial $x^3+x^2-2x-1$ of $2\cos(2\pi/7)$, equal to $-f(-x)$ for the reduced polynomial $f=x^3-x^2-2x+1$ of the entry. The thirteen pure cubic fields here, those with $D=-3f^2$, are named $\mathbb{Q}(\sqrt[3]{m})$ in their comments, with the least $m$ for which that holds; for $m=10$, $17$, $19$, $28$ and $44$ the reduced polynomial is not $x^3-m$. $\mathbb{Q}(\sqrt[3]{2})$ is $D=-108$, the pure cubic field of smallest $|D|$.
(10)
The regulators of real quadratic fields, a single logarithm each, are in the table for degree two; an imaginary quadratic field, like $\mathbb{Q}$, has regulator $1$. A cubic field has unit rank $1$ or $2$ and never $0$, which is why both signatures are listed here. For $D<0$ the regulator $\log\sigma(\varepsilon)$ is, like that of a real quadratic field, transcendental by the Lindemann–Weierstrass theorem, $\sigma(\varepsilon)$ being an algebraic number other than $1$; for $D>0$ the regulator is a $2\times2$ determinant of logarithms of algebraic numbers, about which that theorem says nothing.
(11)
The regulator of an elliptic curve, held in the tables of regulators of elliptic curves of rank 1, rank 2 and rank 3, is the determinant of the canonical height pairing on the Mordell–Weil group, a different object under the same name.
Programs
(P1)
PARI/GP
default(realprecision, 60); bnfinit(x^3 - x^2 - 2*x + 1, 1).reg    \\ D = 49: 0.525454682122572388338826045448324509541055046011794785067649
(P2)
Sage
K.<a> = NumberField(x^3 - x^2 - 2*x + 1)     # D = 49
K.units(proof=True)                          # (-a^2 + a + 1, -a), a fundamental pair
K.regulator(proof=True)                      # 0.525454682122572, a 53-bit float
References
[1]
H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer, 1993; Theorem 6.4.2 (Hunter) and Tables B.3, B.4
Links
Similar tables
Regulators of real quadratic fields —   the same regulator for degree two, $R_K=\log\varepsilon_K$; its entry comments carry $\varepsilon_K$ the same way
Residues of Dedekind zeta functions of quadratic fields —   the other side of the class number formula, for degree two; its entry comments carry $h_K$
Values of Dirichlet $L$-functions at positive integers —   $R_K=\sqrt{D}\,|L(1,\chi)|^2/(4h_K)$ for the seven cyclic fields, $\chi$ a cubic character of conductor $\sqrt{D}$
Regulators of elliptic curves over $\mathbb{Q}$ of rank 1 —   a different regulator, of the height pairing on an elliptic curve; likewise the table for rank 2 and the one for rank 3
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every cubic field with $|D|\leq 3000$, the 419 complex and the 96 totally real ones)
How they were obtained:

The fundamental units are PARI's, from bnfinit and certified by bnfcertify on every field, which removes the generalised Riemann hypothesis that bnfinit assumes.

more

Each unit is taken exactly, as a polynomial in $a$ with rational coefficients, and checked to be an algebraic integer of norm $\pm1$; the real roots of the reduced polynomial are isolated in interval arithmetic at 461 bits, the unit is evaluated at them by Horner's rule in balls, and the logarithms and the determinant are arb's, so the digits written are those the ball supports (the worst entry, $D=1772$, supports 133). PARI's floating regulator is not used for the value. The enumeration rests on Hunter's theorem [1]: every cubic field with $|D|\leq 3000$ is generated by an algebraic integer of trace $0$ or $1$ whose minimal polynomial $x^3+a_2x^2+a_1x+a_0$ has $|a_1|\leq 18$ and $|a_0|\leq 43$, the bound using Hermite's constant $\gamma_2=2/\sqrt3$; the fields found in that box, 419 complex and 96 totally real, are exactly the terms of OEIS A023679 [8] and A006832 [9] up to 3000 counted with multiplicity, and the totally real ones are also those of Sage's enumerate_totallyreal_fields_prim. Outside the generator, when this was written, the class number formula from $h_K$ and each value was compared on all 515 fields with the residue that PARI's lfunrootres computes from the Dedekind zeta function, a computation sharing no code with bnfinit; 21 values with the twelve digits on their LMFDB pages [5] [6]; the entry for $D=-23$ with the 34 digits of OEIS A202539 and with the logarithm of A060006; the four cyclic fields of conductor at most $29$ with $|L(1,\chi)|^2$ for both cubic characters from the stored table of Dirichlet $L$-values; and the pairs of units that Wikipedia and the LMFDB give for $D=49$ were shown to have the same regulator as the pair here. The controls that must fail failed.



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