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Mahler measures of the Hessian family

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Numbers
$t$ 
$N(t)$
-1:
0.4549624460247022157899646290452669704837710430448280179561721677447580407691966814707910045615975207
comment: The negative real boundary of the torus-zero region, with $k=-1$.
0:
0.3230659472194505140936365107238063940722418407805870161308684703610151128072698420837876090893713921
comment: The Fermat specialization gives $N(0)=\mu_3$, the three-step Mahler measure.
27:
0.9691978416583515422809095321714191822167255223417610483926054110830453384218095262513628272681141762
comment: The cubic factors into three linear factors, each of Mahler measure $\mu_3$, so $N(27)=3\mu_3$.
-216:
1.800714521389232519501185405743840291737296186591797905532716386162363170058159818083129625765602197
comment: At $t=t_Q([1,1,1])$, $C_t$ has CM order discriminant $\Delta=-3$. The parameter is rational. [2], Section 1.
24:
0.9017689717714497038711883043277513920447874379412718951342713018471165138878658154816614965952738831
comment: At $t=t_Q([9,3,1])$, $C_t$ has CM order discriminant $\Delta=-27$. The parameter is rational. [2], Section 1. An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
54:
1.285783612394876596075276669356046539933373792348389251008786266766574524229329991694179167604592026
comment: At $t=t_Q([3,0,1])$, $C_t$ has CM order discriminant $\Delta=-12$. The parameter is rational. [2], Section 1.
$108+54\sqrt{2}$:
1.727655340123065766541154958654423253125575351078219975610305918610812048989139033331494265061730081
comment: At $t=t_Q([3,0,2])$, $C_t$ has CM order discriminant $\Delta=-24$. The parameter is real quadratic. [2], Table 1, row 13.
$108-54\sqrt{2}$:
1.059905935890393452450890400849350487614758432820326968317339530385018099053017691335844399986334278
comment: At $t=t_Q([6,0,1])$, $C_t$ has CM order discriminant $\Delta=-24$. The parameter is real quadratic. [2], Table 1, row 13.
$18+6\sqrt{3}$:
1.003029834811605129839867869536644140453242834722302446325158096488593803194804733989674878283400100
comment: At $t=t_Q([9,0,1])$, $C_t$ has CM order discriminant $\Delta=-36$. The parameter is real quadratic. [2], Table 1, row 14.
$18-6\sqrt{3}$:
0.5920965469947062494295412092683636467226951844411848086153302840833862418377243337219524125987214725
comment: At $t=t_Q([9,6,2])$, $C_t$ has CM order discriminant $\Delta=-36$. The parameter is real quadratic. [2], Table 1, row 14.
$270+162\sqrt{3}$:
2.099981744178376731791878867270182892750392183773062011256597026657887476581657140413734511434010101
comment: At $t=t_Q([1,0,1])$, $C_t$ has CM order discriminant $\Delta=-4$. The parameter is real quadratic. [2], Table 1, row 15.
$270-162\sqrt{3}$:
0.9091077602564386577277247413397495286093363203938453277188772628078939330027570615552901234161901998
comment: At $t=t_Q([2,2,1])$, $C_t$ has CM order discriminant $\Delta=-4$. The parameter is real quadratic. [2], Table 1, row 15.
$729+405\sqrt{3}$:
2.420516385163434449629413366858139565791557279776831965552730062644549948187330118305482191174311361
comment: At $t=t_Q([3,0,4])$, $C_t$ has CM order discriminant $\Delta=-48$. The parameter is real quadratic. [2], Table 1, row 16.
$729-405\sqrt{3}$:
0.9836819331422391110329967256481395117829931825084961780791013249893763236866702615284268795348466448
comment: At $t=t_Q([12,0,1])$, $C_t$ has CM order discriminant $\Delta=-48$. The parameter is real quadratic. [2], Table 1, row 16.
$\frac{-27}{2}-\frac{27}{2}\sqrt{5}$:
1.298605737392881591792249284500453734684030523082832218706201803316600428159594233358650806356989239
comment: At $t=t_Q([3,-3,2])$, $C_t$ has CM order discriminant $\Delta=-15$. The parameter is real quadratic. [2], Table 1, row 17.
$\frac{-27}{2}+\frac{27}{2}\sqrt{5}$:
0.7680434591650134888193730218685892439272060749216783350865008980102679556463861436772083721875979691
comment: At $t=t_Q([6,3,1])$, $C_t$ has CM order discriminant $\Delta=-15$. The parameter is real quadratic. [2], Table 1, row 17.
$\frac{3375}{2}+\frac{1485}{2}\sqrt{5}$:
2.704752199209194058624746451594970438569777621703634521174737755763400434983163804439756233944022815
comment: At $t=t_Q([3,0,5])$, $C_t$ has CM order discriminant $\Delta=-60$. The parameter is real quadratic. [2], Table 1, row 18.
$\frac{3375}{2}-\frac{1485}{2}\sqrt{5}$:
0.9759835641225889666715056367182648270788807964689219674191993559260016921676682201912869426668562285
comment: At $t=t_Q([15,0,1])$, $C_t$ has CM order discriminant $\Delta=-60$. The parameter is real quadratic. [2], Table 1, row 18.
$-4320-1944\sqrt{5}$:
3.022653334601145585262377400398605184457541948537194128053826472344298815730259755066552833912772892
comment: At $t=t_Q([3,-3,7])$, $C_t$ has CM order discriminant $\Delta=-75$. The parameter is real quadratic. [2], Table 1, row 19.
$-4320+1944\sqrt{5}$:
0.9663270874013787041406197662486857338628364712017570767216309787834215036807874658818125360598719301
comment: At $t=t_Q([21,3,1])$, $C_t$ has CM order discriminant $\Delta=-75$. The parameter is real quadratic. [2], Table 1, row 19.
$32+2\sqrt{6}$:
1.130631303161317244951209819006588591957195612395167473278322660001297126566378926469503558130560769
comment: At $t=t_Q([9,0,2])$, $C_t$ has CM order discriminant $\Delta=-72$. The parameter is real quadratic. [2], Table 1, row 20.
$32-2\sqrt{6}$:
0.9725915934916164598735756838801957322347906660861116463548964739468381202515279599165171961333108118
comment: At $t=t_Q([18,0,1])$, $C_t$ has CM order discriminant $\Delta=-72$. The parameter is real quadratic. [2], Table 1, row 20.
$3672+1458\sqrt{6}$:
2.962336905206945724844014805642105291978386650950761319767718364651972668738726815067680701015745399
comment: At $t=t_Q([1,0,2])$, $C_t$ has CM order discriminant $\Delta=-8$. The parameter is real quadratic. [2], Table 1, row 21.
$3672-1458\sqrt{6}$:
1.515626182706226909737536716382730980576381145937482192642543332541757529542645955492204418890684715
comment: At $t=t_Q([2,0,1])$, $C_t$ has CM order discriminant $\Delta=-8$. The parameter is real quadratic. [2], Table 1, row 21.
$-864-216\sqrt{17}$:
2.491132013779120844135437254110611891211852906172703926477410677344981893069857893797590982354393743
comment: At $t=t_Q([3,-3,5])$, $C_t$ has CM order discriminant $\Delta=-51$. The parameter is real quadratic. [2], Table 1, row 22.
$-864+216\sqrt{17}$:
0.9573868395346952929596114534383981915360180479862203843028483173350888318620065904514728450276440335
comment: At $t=t_Q([15,3,1])$, $C_t$ has CM order discriminant $\Delta=-51$. The parameter is real quadratic. [2], Table 1, row 22.
$-163296-35640\sqrt{21}$:
4.232189331912333939917662110313971819477043143599539860830536987919254536131624970969316424092532517
comment: At $t=t_Q([3,-3,13])$, $C_t$ has CM order discriminant $\Delta=-147$. The parameter is real quadratic. [2], Table 1, row 23.
$-163296+35640\sqrt{21}$:
0.9690938348865934061032467741226386558059164917456072943424975934076455079376955618820055877776451835
comment: At $t=t_Q([39,3,1])$, $C_t$ has CM order discriminant $\Delta=-147$. The parameter is real quadratic. [2], Table 1, row 23.
$4-4\sqrt{33}$:
1.060313853042720113357474149455609607271032448715052961574037589506501775789646915487615814676119601
comment: At $t=t_Q([9,-9,5])$, $C_t$ has CM order discriminant $\Delta=-99$. The parameter is real quadratic. [2], Table 1, row 24.
$4+4\sqrt{33}$:
0.9683531251520881988303042692441578703441717364098282766471597927345970196613304981384267750852166606
comment: At $t=t_Q([27,3,1])$, $C_t$ has CM order discriminant $\Delta=-99$. The parameter is real quadratic. [2], Table 1, row 24.
$-16740-2916\sqrt{33}$:
3.473071818869579295240188145912393311569990624146763296335661262207025114006047979393497341840326488
comment: At $t=t_Q([1,-1,3])$, $C_t$ has CM order discriminant $\Delta=-11$. The parameter is real quadratic. [2], Table 1, row 25.
$-16740+2916\sqrt{33}$:
0.6650239619828272423587108944358792115552304440499644641099408088759290276016741318065889380791871040
comment: At $t=t_Q([5,3,1])$, $C_t$ has CM order discriminant $\Delta=-11$. The parameter is real quadratic. [2], Table 1, row 25.
$-55296-8640\sqrt{41}$:
3.871300440182100051356811451853387873673770608739135498460201822237140445768628320239310401861257802
comment: At $t=t_Q([3,-3,11])$, $C_t$ has CM order discriminant $\Delta=-123$. The parameter is real quadratic. [2], Table 1, row 26.
$-55296+8640\sqrt{41}$:
0.9689151101674892016309712836393120891491166353520058919461376976098160939137379280693304222179265571
comment: At $t=t_Q([33,3,1])$, $C_t$ has CM order discriminant $\Delta=-123$. The parameter is real quadratic. [2], Table 1, row 26.
$-13500000-1431000\sqrt{89}$:
5.703782882063013817942765984840551372015462014636724568739699272076956711941714749660016641013224744
comment: At $t=t_Q([3,-3,23])$, $C_t$ has CM order discriminant $\Delta=-267$. The parameter is real quadratic. [2], Table 1, row 27.
$-13500000+1431000\sqrt{89}$:
0.9691961774780599931093656328095021079956638816652182588008551473090827020184711084142924091287484464
comment: At $t=t_Q([69,3,1])$, $C_t$ has CM order discriminant $\Delta=-267$. The parameter is real quadratic. [2], Table 1, row 27.
1:
0.4000616683584389676514859682701250565690975709056252749701394669026734354517382831375364640440484856
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
2:
0.4412220238746347811626379432554650206815552956251371234406720748405765510231487751990803991896741625
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
3:
0.4746769034641825745524505479721470036522980499159844517435215187250672725608913460082946711339200025
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
4:
0.5039849021054002429296863265563046727036465144309280974825770830531789617008778341840211358165370694
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
5:
0.5305995603311479891158550823856709197799210072040720802564228541352044510857577404290612269965332104
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
6:
0.5552934972313736558003500725787058902497905770709039908676992085639708447550904592632475924725331249
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
7:
0.5785392008618616326530185083673879831253512660684869432845349677205648993313807019423813607613584399
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
8:
0.6006528136218932204191858980413420159158772799329260545203919838947376403277680859118224573898089453
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
9:
0.6218598877162198858621959208579691340765709577090789596882792263303116739236013161148887767874243984
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
10:
0.6423294984869220468144704711588143529381128320134224020675285531779212679330907053484688959117621058
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
11:
0.6621936372947107984963802268891662784510988755430972796431849845906529065838999425779788721589225415
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
12:
0.6815590543760703664936309632520336201191752171408341767494056748198417254719246707338798675909413546
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
13:
0.7005149438873445905665882964150501594412633701180610248121121592034541180809500918806509066563930474
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
14:
0.7191382229010127377843158302417064347002900030395475449739431253340166066626584055178911256548218916
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
15:
0.7374973820762809100303644194563140833855513765555397683587086819294348390058955939109280055053597642
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
16:
0.7556555006518545062469755896604658627174855650463927197997606024101831361728182919339405837048730997
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
17:
0.7736728268224347357277328098288173800159024514412368326800011254288473157893190718154335787324290273
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
18:
0.7916092444978240769791996962407093006688159675484850971932828457392411982769519683222180975932418134
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
19:
0.8095269518192670650752950182675284490863250430492534207284984557620005255576007889698518374872537127
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
20:
0.8274937784929041469452709092705575179820939536361813320000819757487506637151328068061129494534936572
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
21:
0.8455878372064046280450891376573168862092111837308456939966792343423664756958278910261610108121076431
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
22:
0.8639048389629331752888647777064379555477669035082326238459023379469364358941904419545434771704730501
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
23:
0.8825709932201857978109812745939654659105022177387278800687569806129826006692325299042073729697931377
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
25:
0.9218005938001589722445865664868523434084286832596097765958268014400678863579396382491026463284130193
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
26:
0.9432933462372278194081835106072776607512989984570862491865255988343503237367790986511019318892305219
comment: An ordinary companion at an integer parameter of the modified-period arithmetic family in [1], Table 2; those proposed $L$-value identities are for $\widetilde N$, not $N$.
-27:
1.158250457681781571237335414497831183418952805988383141004311327719647095132045174899788937787688867
comment: An exterior Hessian argument with a proved $L$-value evaluation listed in [1], Table 1.
-8:
0.8416338457402896019772381357839460632391016163179668167516686265081844755814218506043468794092021714
comment: An exterior Hessian argument with a proved $L$-value evaluation listed in [1], Table 1.
32:
1.065512372801819887139359733201972124656099195833271253128995525440372378332151898579444938349705097
comment: An exterior Hessian argument with a proved $L$-value evaluation listed in [1], Table 1. (5) relates this value to $g(4)$.
125:
1.592368561086457755264876201658434396693198650656898062846602587106653142692188385147768515965591322
comment: An exterior Hessian argument with a proved $L$-value evaluation listed in [1], Table 1.
$t_Q([27,0,1])$:
0.9697514661623911906416788164679917924955989603046442730479919555310798230366478814073678010400371571
comment: At $t=t_Q([27,0,1])$, $C_t$ has CM order discriminant $\Delta=-108$. The parameter is real cubic; its other two algebraic conjugates are nonreal. [5], entry ν-C-1-1.
$t_Q([1,0,3])$:
3.627655062481541440859715335115863110855144341795368345919899186204492675914361027286616871875367371
comment: At $t=t_Q([1,0,3])$, $C_t$ has CM order discriminant $\Delta=-12$. The parameter is real cubic; its other two algebraic conjugates are nonreal. [5], entry ν-C-2-1.
$t_Q([63,3,1])$:
0.9691943441846044857910358057491044519282411839841496215384008926977907924778436616436467064460282716
comment: At $t=t_Q([63,3,1])$, $C_t$ has CM order discriminant $\Delta=-243$. The parameter is real cubic; its other two algebraic conjugates are nonreal. [5], entry ν-C-3-1.
$t_Q([1,-1,7])$:
5.441397848576779491748211851381930511415207882349066292705023132748923904780156295147453862633512313
comment: At $t=t_Q([1,-1,7])$, $C_t$ has CM order discriminant $\Delta=-27$. The parameter is real cubic; its other two algebraic conjugates are nonreal. [5], entry ν-C-4-1.
$t_Q([27,24,8])$:
0.3714352326609706011606526480942424294862223597919193271253944047318413651893282943868325177692571872
comment: At $t=t_Q([27,24,8])$, $C_t$ has CM order discriminant $\Delta=-288$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 52.
$t_Q([27,12,4])$:
0.8191410808197767512945774701309390035872581970791722291891929118360030891759631232118390349123628313
comment: At $t=t_Q([27,12,4])$, $C_t$ has CM order discriminant $\Delta=-288$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 52.
$t_Q([72,0,1])$:
0.9691987330012687390572414545769306005451837065372770096881101861766544641075850419108247592991675238
comment: At $t=t_Q([72,0,1])$, $C_t$ has CM order discriminant $\Delta=-288$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 52.
$t_Q([9,0,8])$:
1.982607155857683377084200425898804048510715719668612939716272313860975248593122601500635173831068846
comment: At $t=t_Q([9,0,8])$, $C_t$ has CM order discriminant $\Delta=-288$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 52.
$t_Q([12,12,5])$:
0.5976037439994184941613912834409628810045779023029709331244942957899692410265323172327164654843034347
comment: At $t=t_Q([12,12,5])$, $C_t$ has CM order discriminant $\Delta=-96$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 53.
$t_Q([15,12,4])$:
0.4499629340377789199985566486773165455870251229979363749039957271221119611724730332184030572550895882
comment: At $t=t_Q([15,12,4])$, $C_t$ has CM order discriminant $\Delta=-96$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 53.
$t_Q([24,0,1])$:
0.9701754486658952145481182971059715091800916010303237515006691651302613501460945224662289399051589726
comment: At $t=t_Q([24,0,1])$, $C_t$ has CM order discriminant $\Delta=-96$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 53.
$t_Q([3,0,8])$:
3.420237850129504623117873280126791563998031194178978914136375791003711743523483554281521830327913978
comment: At $t=t_Q([3,0,8])$, $C_t$ has CM order discriminant $\Delta=-96$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 53.
$t_Q([12,-12,7])$:
1.126266836154737727500399861191453555034625498279783625856717635713341119931248123594472544397031605
comment: At $t=t_Q([12,-12,7])$, $C_t$ has CM order discriminant $\Delta=-192$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 54.
$t_Q([21,12,4])$:
0.6955380773122398074171449003851788314459634416381621351016373951629083776466065427657282207542780865
comment: At $t=t_Q([21,12,4])$, $C_t$ has CM order discriminant $\Delta=-192$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 54.
$t_Q([48,0,1])$:
0.9692170487671929985878304465659260925627011232486181794320653954096905737356577957809782436343580129
comment: At $t=t_Q([48,0,1])$, $C_t$ has CM order discriminant $\Delta=-192$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 54.
$t_Q([3,0,16])$:
4.836799802019958836231958315560959396171335564979334386761847045206782744361284235102549142762278858
comment: At $t=t_Q([3,0,16])$, $C_t$ has CM order discriminant $\Delta=-192$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 54.
$t_Q([4,-4,3])$:
1.445051051596592112195999438778994207589131606980650468982120226086533991671823354984742971598535914
comment: At $t=t_Q([4,-4,3])$, $C_t$ has CM order discriminant $\Delta=-32$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 55.
$t_Q([8,8,3])$:
0.3944654674440439921468470156836359783271233314612748294057905386285850959333004661340704404705851601
comment: At $t=t_Q([8,8,3])$, $C_t$ has CM order discriminant $\Delta=-32$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 55.
$t_Q([8,0,1])$:
1.015318970924568598377298652544660955997851836455531355955075275633008433758426381100152233986154978
comment: At $t=t_Q([8,0,1])$, $C_t$ has CM order discriminant $\Delta=-32$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 55.
$t_Q([1,0,8])$:
5.923843974961828098232790286044020532050336352885208310463909326292104763715324877839994746768228413
comment: At $t=t_Q([1,0,8])$, $C_t$ has CM order discriminant $\Delta=-32$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 55.
$t_Q([30,0,1])$:
0.9695203563827805208113186919032829080057766798430092763384899886253790164569330472302880286571434606
comment: At $t=t_Q([30,0,1])$, $C_t$ has CM order discriminant $\Delta=-120$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 56.
$t_Q([15,0,2])$:
1.023393477414518999950842650044643117096476921468096480909346969194602019283729201190331483170891222
comment: At $t=t_Q([15,0,2])$, $C_t$ has CM order discriminant $\Delta=-120$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 56.
$t_Q([6,0,5])$:
1.921551037617664465057165355958835781502581360133196425404584785034288776105776740911484153524812694
comment: At $t=t_Q([6,0,5])$, $C_t$ has CM order discriminant $\Delta=-120$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 56.
$t_Q([3,0,10])$:
3.823856075538795315431555320831423663374844693711124888389840411958769504299510508280504629877285088
comment: At $t=t_Q([3,0,10])$, $C_t$ has CM order discriminant $\Delta=-120$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 56.
$t_Q([18,18,7])$:
0.4508050762055176193886207308698376177848410158216005361439617800227002481291734012442864883144245792
comment: At $t=t_Q([18,18,7])$, $C_t$ has CM order discriminant $\Delta=-180$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 57.
$t_Q([27,6,2])$:
0.9516414718109270831829565697919919460077355937811487689660748969538376197499206458360313269256907862
comment: At $t=t_Q([27,6,2])$, $C_t$ has CM order discriminant $\Delta=-180$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 57.
$t_Q([45,0,1])$:
0.9692273859121797723904565667935432694516254310997324676111399612741400638341064419763688774988030881
comment: At $t=t_Q([45,0,1])$, $C_t$ has CM order discriminant $\Delta=-180$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 57.
$t_Q([9,0,5])$:
1.588433663759590681545845605377701045721877513030403992846606486609014914525028382245184989494270238
comment: At $t=t_Q([9,0,5])$, $C_t$ has CM order discriminant $\Delta=-180$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 57.
$t_Q([2,-2,3])$:
2.338932752023205740570460985528331204677195700831250526888090259127201869653089659056610800094777201
comment: At $t=t_Q([2,-2,3])$, $C_t$ has CM order discriminant $\Delta=-20$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 58.
$t_Q([7,6,2])$:
0.3969057399336508538575866133204748079010932481599922585441782705239030401114150040847198771547961013
comment: At $t=t_Q([7,6,2])$, $C_t$ has CM order discriminant $\Delta=-20$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 58.
$t_Q([5,0,1])$:
1.101292020238027265712968255893376524749583053305359254406057942391763073822154790746512838271897847
comment: At $t=t_Q([5,0,1])$, $C_t$ has CM order discriminant $\Delta=-20$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 58.
$t_Q([1,0,5])$:
4.683212194494176059630903572689075485989499294201397876101761552428274345789178591393900301406460097
comment: At $t=t_Q([1,0,5])$, $C_t$ has CM order discriminant $\Delta=-20$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 58.
$t_Q([60,0,1])$:
0.9692016886905758535324775427985884129838553576917899755904600526138246467999905692178459563397468769
comment: At $t=t_Q([60,0,1])$, $C_t$ has CM order discriminant $\Delta=-240$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 59.
$t_Q([15,0,4])$:
1.191732597409299705837124701632401610566517036469891745808594009407415068907725160494073367761618825
comment: At $t=t_Q([15,0,4])$, $C_t$ has CM order discriminant $\Delta=-240$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 59.
$t_Q([12,0,5])$:
1.402548007760183028841355935350002848942775836412656956840590044070960151461050440680045190041431269
comment: At $t=t_Q([12,0,5])$, $C_t$ has CM order discriminant $\Delta=-240$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 59.
$t_Q([3,0,20])$:
5.407705171390323398253922287317683949668069612303196369535502557338880942285068043678982349115244477
comment: At $t=t_Q([3,0,20])$, $C_t$ has CM order discriminant $\Delta=-240$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 59.
$t_Q([6,-6,5])$:
1.574597342040760282978335942083855255825519638374030808538938872960691473353583727588198037620749552
comment: At $t=t_Q([6,-6,5])$, $C_t$ has CM order discriminant $\Delta=-84$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 60.
$t_Q([15,6,2])$:
0.8556727428348770820454539105765656485325851239167934190252639452848639907330832079691610788992304527
comment: At $t=t_Q([15,6,2])$, $C_t$ has CM order discriminant $\Delta=-84$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 60.
$t_Q([21,0,1])$:
0.9709828465236518118168422365616639946515817585518308190048365467264502407071176705549815992391440833
comment: At $t=t_Q([21,0,1])$, $C_t$ has CM order discriminant $\Delta=-84$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 60.
$t_Q([3,0,7])$:
3.199444992386936893977882644674372409693593537687525873483527203297951137387562466378163442544073839
comment: At $t=t_Q([3,0,7])$, $C_t$ has CM order discriminant $\Delta=-84$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 60.
$t_Q([6,-6,7])$:
1.997883358463880371738392428870945945941853223380090930640762485840232688506988998010942113914503197
comment: At $t=t_Q([6,-6,7])$, $C_t$ has CM order discriminant $\Delta=-132$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 61.
$t_Q([21,6,2])$:
0.9274658409121505117611286498015907564581211783950663799510014172031937257641315565152135291137466372
comment: At $t=t_Q([21,6,2])$, $C_t$ has CM order discriminant $\Delta=-132$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 61.
$t_Q([33,0,1])$:
0.9693903519320145123073236314858511633913222462887125282307198321681605584986825316816042027745558575
comment: At $t=t_Q([33,0,1])$, $C_t$ has CM order discriminant $\Delta=-132$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 61.
$t_Q([3,0,11])$:
4.010479153740066804309362289890437497506681259073055672190487822865967257847428685372390257535435697
comment: At $t=t_Q([3,0,11])$, $C_t$ has CM order discriminant $\Delta=-132$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 61.
$t_Q([78,0,1])$:
0.9691982897480796273048817680589864987120316771211150601848524771546434857629930853725752506006998940
comment: At $t=t_Q([78,0,1])$, $C_t$ has CM order discriminant $\Delta=-312$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 62.
$t_Q([39,0,2])$:
0.9716453949649697223117507964299355477208841015832747779602642262757570732260668566098017843215160151
comment: At $t=t_Q([39,0,2])$, $C_t$ has CM order discriminant $\Delta=-312$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 62.
$t_Q([6,0,13])$:
3.083154815023504258984928092678815051524960795564837071768640224435287796660252103833293311762274239
comment: At $t=t_Q([6,0,13])$, $C_t$ has CM order discriminant $\Delta=-312$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 62.
$t_Q([3,0,26])$:
6.165732262438452755359365609479396695713648174106773132027946645535817723707274968953791298726723257
comment: At $t=t_Q([3,0,26])$, $C_t$ has CM order discriminant $\Delta=-312$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 62.
$t_Q([3,-3,17])$:
4.874437988547638207754168678597137563619608669978955222802279600145507879469995352602964070954645477
comment: At $t=t_Q([3,-3,17])$, $C_t$ has CM order discriminant $\Delta=-195$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 63.
$t_Q([15,15,7])$:
0.8008039767495067550422335662686392816526207271381682794959679698975738741648863868326300551717200008
comment: At $t=t_Q([15,15,7])$, $C_t$ has CM order discriminant $\Delta=-195$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 63.
$t_Q([21,15,5])$:
0.5398452896259738298145318283905351507586034699110214468156406354331290041755601575226562772195402750
comment: At $t=t_Q([21,15,5])$, $C_t$ has CM order discriminant $\Delta=-195$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 63.
$t_Q([51,3,1])$:
0.9691805620923296489415083587313295562692049365573074809874601208990448907698351364763299894505733366
comment: At $t=t_Q([51,3,1])$, $C_t$ has CM order discriminant $\Delta=-195$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 63.
$t_Q([102,0,1])$:
0.9691978774426628394362380068183076220487755961916174766530040792489170092139320080387988955492122672
comment: At $t=t_Q([102,0,1])$, $C_t$ has CM order discriminant $\Delta=-408$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 64.
$t_Q([51,0,2])$:
0.9699307152709286397076965341120104326418991339447162633853317486875663083560831102788958665424543607
comment: At $t=t_Q([51,0,2])$, $C_t$ has CM order discriminant $\Delta=-408$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 64.
$t_Q([6,0,17])$:
3.525468827131238844347431428838752907293174819901106748375694046391016508374509831811477790782051831
comment: At $t=t_Q([6,0,17])$, $C_t$ has CM order discriminant $\Delta=-408$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 64.
$t_Q([3,0,34])$:
7.050784561787117455975340205548955850297551865912797310277737979200949041381141783743441112993216130
comment: At $t=t_Q([3,0,34])$, $C_t$ has CM order discriminant $\Delta=-408$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 64.
$t_Q([42,0,1])$:
0.9692439097646217626438974576514063235988829757303613841814570481260450031536003192772076128867869193
comment: At $t=t_Q([42,0,1])$, $C_t$ has CM order discriminant $\Delta=-168$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 65.
$t_Q([21,0,2])$:
0.9910387812261480110401331674902242103181127854261268743500887019032864388240818893519340117857612092
comment: At $t=t_Q([21,0,2])$, $C_t$ has CM order discriminant $\Delta=-168$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 65.
$t_Q([6,0,7])$:
2.265585877949689953445898436457534335265278837676010023601067782044788155856473143760422838346120223
comment: At $t=t_Q([6,0,7])$, $C_t$ has CM order discriminant $\Delta=-168$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 65.
$t_Q([3,0,14])$:
4.524414348650068297849822093313192343247604045308630037307301220483047303101460304021022496126591080
comment: At $t=t_Q([3,0,14])$, $C_t$ has CM order discriminant $\Delta=-168$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 65.
$t_Q([6,-6,11])$:
2.634283375173639881159524752527691351673173287456687642250960503165523651902845351268324167014830112
comment: At $t=t_Q([6,-6,11])$, $C_t$ has CM order discriminant $\Delta=-228$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 66.
$t_Q([33,6,2])$:
0.9610955876682968742776913623011760578779203987287290953288786768447126766941369982605985803816048482
comment: At $t=t_Q([33,6,2])$, $C_t$ has CM order discriminant $\Delta=-228$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 66.
$t_Q([57,0,1])$:
0.9692035045863017155602482914056677372110217737058798474929847157572707798906172877902304521007225451
comment: At $t=t_Q([57,0,1])$, $C_t$ has CM order discriminant $\Delta=-228$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 66.
$t_Q([3,0,19])$:
5.270779162333447575253820393125714825448925144488802742843078856703147038070362121775560063730421865
comment: At $t=t_Q([3,0,19])$, $C_t$ has CM order discriminant $\Delta=-228$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 66.
$t_Q([9,-9,11])$:
2.058965755516038131294077197524792985574350092906331049990660904879710472734729380264053586069285947
comment: At $t=t_Q([9,-9,11])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([27,21,7])$:
0.4723942110159338522884311794178486657114362948081133845955470704163364286683646551698937471791480997
comment: At $t=t_Q([27,21,7])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([27,15,5])$:
0.7121749478283680937175508120439359922869886303751626664835754377536556832335427782485206421463187658
comment: At $t=t_Q([27,15,5])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([81,3,1])$:
0.9691974297471212093615681137600823621034616369748870368656868060543381745080284498044432980035750089
comment: At $t=t_Q([81,3,1])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([1,-1,9])$:
6.195304236114707304561082934956885451270413815295372923174875304042908084930251316239236612689580264
comment: At $t=t_Q([1,-1,9])$, $C_t$ has CM order discriminant $\Delta=-35$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 68.
$t_Q([5,-5,3])$:
1.163485278784124955110130037740678849617964876931503095135383437716187045602566703346922804306860276
comment: At $t=t_Q([5,-5,3])$, $C_t$ has CM order discriminant $\Delta=-35$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 68.
$t_Q([7,7,3])$:
0.6518566312033200408229089120309821236209507359647395109278192048467552019698099873926064153633013353
comment: At $t=t_Q([7,7,3])$, $C_t$ has CM order discriminant $\Delta=-35$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 68.
$t_Q([11,3,1])$:
0.9333585838450204515137490183193413228495640849877032249891875472420788197719682664577115132882119923
comment: At $t=t_Q([11,3,1])$, $C_t$ has CM order discriminant $\Delta=-35$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 68.
$t_Q([3,-3,37])$:
7.280345529478866051604538798501844675456646187373109703078436782449573435906293639467169596996635064
comment: At $t=t_Q([3,-3,37])$, $C_t$ has CM order discriminant $\Delta=-435$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 69.
$t_Q([15,-15,11])$:
1.417322675428314344399475124628863011612326435852396816850868490435803850790739917206607155244934983
comment: At $t=t_Q([15,-15,11])$, $C_t$ has CM order discriminant $\Delta=-435$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 69.
$t_Q([33,15,5])$:
0.8102679189372928163500802731559223988109981512096780715159162631231985938674654832584413774230565424
comment: At $t=t_Q([33,15,5])$, $C_t$ has CM order discriminant $\Delta=-435$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 69.
$t_Q([111,3,1])$:
0.9691978231274427104186883814216461930806149829816727033498025291344044614031189862566549201659067533
comment: At $t=t_Q([111,3,1])$, $C_t$ has CM order discriminant $\Delta=-435$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 69.
$t_Q([6,-6,17])$:
3.366145760503393547777679825248803903691938345964798566565569023934183397813374512387052261870789652
comment: At $t=t_Q([6,-6,17])$, $C_t$ has CM order discriminant $\Delta=-372$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 70.
$t_Q([51,6,2])$:
0.9680667040331016053063094198706655567483691508438169313831308417820412368834270061575334518371353040
comment: At $t=t_Q([51,6,2])$, $C_t$ has CM order discriminant $\Delta=-372$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 70.
$t_Q([93,0,1])$:
0.9691979306189220344528007136343700809400676660833819034098519746339785288413364049463745236620577624
comment: At $t=t_Q([93,0,1])$, $C_t$ has CM order discriminant $\Delta=-372$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 70.
$t_Q([3,0,31])$:
6.732538312746031172568574423152075563214757635437641177348932178957656357806969904931916720953234706
comment: At $t=t_Q([3,0,31])$, $C_t$ has CM order discriminant $\Delta=-372$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 70.
$t_Q([3,-3,47])$:
8.223446187073670145804606056280279784024633695438865947270630404298725517256334159836475969834467262
comment: At $t=t_Q([3,-3,47])$, $C_t$ has CM order discriminant $\Delta=-555$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 71.
$t_Q([15,-15,13])$:
1.622864291607015347006296731836415644691758040945222503998320749155335805146608828343438919648757194
comment: At $t=t_Q([15,-15,13])$, $C_t$ has CM order discriminant $\Delta=-555$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 71.
$t_Q([39,15,5])$:
0.8672884796006425327155351192626246498535633033475736361580853622024491308075295773881566877063523884
comment: At $t=t_Q([39,15,5])$, $C_t$ has CM order discriminant $\Delta=-555$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 71.
$t_Q([141,3,1])$:
0.9691978404284713349891325152468989686980634381721046613589244707419449520611496611054048837041984330
comment: At $t=t_Q([141,3,1])$, $C_t$ has CM order discriminant $\Delta=-555$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 71.
$t_Q([3,-3,67])$:
9.842171482562395499474661182978255162309425501471189450328503688128586881832007883031948362099185415
comment: At $t=t_Q([3,-3,67])$, $C_t$ has CM order discriminant $\Delta=-795$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 72.
$t_Q([15,-15,17])$:
1.960225993692896549198225885452008481208584080049987281021318292958851151754548231349475035829008729
comment: At $t=t_Q([15,-15,17])$, $C_t$ has CM order discriminant $\Delta=-795$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 72.
$t_Q([51,15,5])$:
0.9233948066518742389551737595095614957682364304386067120894436993658438739132180650378359939434633689
comment: At $t=t_Q([51,15,5])$, $C_t$ has CM order discriminant $\Delta=-795$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 72.
$t_Q([201,3,1])$:
0.9691978416469726972748901785642465127684589944782440225576289084685665780033527728340549257353583954
comment: At $t=t_Q([201,3,1])$, $C_t$ has CM order discriminant $\Delta=-795$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 72.
$t_Q([6,-6,31])$:
4.644016421389027133856784934999243244967669598180036578047984927697347576830857167732296087174377780
comment: At $t=t_Q([6,-6,31])$, $C_t$ has CM order discriminant $\Delta=-708$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 73.
$t_Q([93,6,2])$:
0.9691648752313650191004866604460061689785912936850441286631758232488570118045625976328080640519869699
comment: At $t=t_Q([93,6,2])$, $C_t$ has CM order discriminant $\Delta=-708$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 73.
$t_Q([177,0,1])$:
0.9691978417150733830584101575673716593625032290670979301628236037274536658475319424437935672251072843
comment: At $t=t_Q([177,0,1])$, $C_t$ has CM order discriminant $\Delta=-708$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 73.
$t_Q([3,0,59])$:
9.288038182718674446654199326624552424345962160669882969691253446951053991930825501389607253784623943
comment: At $t=t_Q([3,0,59])$, $C_t$ has CM order discriminant $\Delta=-708$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 73.
$t_Q([3,-3,41])$:
7.671511291664557499148997888990801464309606579081297472981502255551954402301966265641144668303672733
comment: At $t=t_Q([3,-3,41])$, $C_t$ has CM order discriminant $\Delta=-483$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 74.
$t_Q([21,-21,11])$:
0.9776163902674831251985142937724852560546623545911409444041107109256174088160606155651651609839914893
comment: At $t=t_Q([21,-21,11])$, $C_t$ has CM order discriminant $\Delta=-483$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 74.
$t_Q([33,21,7])$:
0.6254285842575834686752038467147464326779803059991861057753166135039685826612199741850528505831301781
comment: At $t=t_Q([33,21,7])$, $C_t$ has CM order discriminant $\Delta=-483$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 74.
$t_Q([123,3,1])$:
0.9691978356325891842722270615346113251865526278063338373149926498782586299641609100241027266019519806
comment: At $t=t_Q([123,3,1])$, $C_t$ has CM order discriminant $\Delta=-483$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 74.
$t_Q([3,-3,53])$:
8.740597741704618448215205508044323636831008656964173695827549762695556403020042319837926780729428108
comment: At $t=t_Q([3,-3,53])$, $C_t$ has CM order discriminant $\Delta=-627$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 75.
$t_Q([33,33,13])$:
0.4797620154350908677154675684891498180144532263146276766977198341835409562881779397825603375258297654
comment: At $t=t_Q([33,33,13])$, $C_t$ has CM order discriminant $\Delta=-627$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 75.
$t_Q([39,33,11])$:
0.4064759858250428378663040483245267376735297149833819724496511848888272527607611004435577470035957678
comment: At $t=t_Q([39,33,11])$, $C_t$ has CM order discriminant $\Delta=-627$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 75.
$t_Q([159,3,1])$:
0.9691978413819376723158730286001118010348629879046299063853307870025690323306470675721616897297809561
comment: At $t=t_Q([159,3,1])$, $C_t$ has CM order discriminant $\Delta=-627$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 75.
$t_Q([1,-1,2])$:
2.769887271616823038689914670737516966749656320577740553126228836373761589858125166647295947377182667
comment: At $t=t_Q([1,-1,2])$, $C_t$ has CM order discriminant $\Delta=-7$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-1-1.
$t_Q([4,3,1])$:
0.5014132991869672042965850315578185693490610454853763313191954308771433907095814656307360386896590888
comment: At $t=t_Q([4,3,1])$, $C_t$ has CM order discriminant $\Delta=-7$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-1-2.
$t_Q([1,-1,5])$:
4.564624911589687460983838361921522639599154683710802202323224279877225629064763767188244203363215742
comment: At $t=t_Q([1,-1,5])$, $C_t$ has CM order discriminant $\Delta=-19$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-2-1.
$t_Q([7,3,1])$:
0.8326721174556971425638235206143407236111961847511707817894292890587366540386031797845045111841993542
comment: At $t=t_Q([7,3,1])$, $C_t$ has CM order discriminant $\Delta=-19$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-2-2.
$t_Q([1,-1,11])$:
6.866933561380447533773482704527684988278718640742837312965656925657938060685075280536612332213719493
comment: At $t=t_Q([1,-1,11])$, $C_t$ has CM order discriminant $\Delta=-43$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-4-1.
$t_Q([13,3,1])$:
0.9490633033321047614040419529103264821835360684295320549146281815440197060693730390414416356483520253
comment: At $t=t_Q([13,3,1])$, $C_t$ has CM order discriminant $\Delta=-43$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-4-2.
$t_Q([9,9,4])$:
0.7179434719662430002720316422012260294090644507751109169863289987998333122567175346011493621078131542
comment: At $t=t_Q([9,9,4])$, $C_t$ has CM order discriminant $\Delta=-63$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-5-1.
$t_Q([18,3,1])$:
0.9635452980876323433629263710976096116975725111540817702691878557940285312493557577717381823598286258
comment: At $t=t_Q([18,3,1])$, $C_t$ has CM order discriminant $\Delta=-63$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-5-2.
$t_Q([1,-1,17])$:
8.571681378364732481095193654966685670662765640339852166540895104250905402300423252540198689507786485
comment: At $t=t_Q([1,-1,17])$, $C_t$ has CM order discriminant $\Delta=-67$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-6-1.
$t_Q([19,3,1])$:
0.9647144173978984032751551236642516691303027504472394362436508125586630865957633884237194299679572444
comment: At $t=t_Q([19,3,1])$, $C_t$ has CM order discriminant $\Delta=-67$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-6-2.
$t_Q([1,-1,41])$:
13.36972333037750657368975172994508960466730605145332209107104447860114994742086001074127665193842060
comment: At $t=t_Q([1,-1,41])$, $C_t$ has CM order discriminant $\Delta=-163$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-7-1.
$t_Q([43,3,1])$:
0.9691421746245128665228794758006102395074006063830091240101874383099437522510680157603882244616648244
comment: At $t=t_Q([43,3,1])$, $C_t$ has CM order discriminant $\Delta=-163$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-7-2.
$t_Q([9,-9,7])$:
1.489812497994892449060638219888913353821327961733312672917685790727328537247274773236176703583198206
comment: At $t=t_Q([9,-9,7])$, $C_t$ has CM order discriminant $\Delta=-171$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-8-1.
$t_Q([45,3,1])$:
0.9691566776440147113024766746922122017333552181335340462418852813779334705128462895559908222029130865
comment: At $t=t_Q([45,3,1])$, $C_t$ has CM order discriminant $\Delta=-171$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-8-2.
$t_Q([9,-9,13])$:
2.285847971871183734965403576173003462500130757810292528969351283799816013624639334260540278922989099
comment: At $t=t_Q([9,-9,13])$, $C_t$ has CM order discriminant $\Delta=-387$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-9-1.
$t_Q([99,3,1])$:
0.9691977810855443236154104783475371886009206337293364560775702632056007963867842217288307991674250088
comment: At $t=t_Q([99,3,1])$, $C_t$ has CM order discriminant $\Delta=-387$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-9-2.
$t_Q([36,0,1])$:
0.9693152188510643354895967997728370656952962234045119685575997404213598413248154385624537305120014540
comment: At $t=t_Q([36,0,1])$, $C_t$ has CM order discriminant $\Delta=-144$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-10-1.
$t_Q([9,0,4])$:
1.440726406959554061528594705324615367378937689260329871300817549858923788761188617053709366105795877
comment: At $t=t_Q([9,0,4])$, $C_t$ has CM order discriminant $\Delta=-144$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-10-2.
$t_Q([63,0,1])$:
0.9692004787211484096073158775852349062108344216293978889746569026439816724552252966321900804966822592
comment: At $t=t_Q([63,0,1])$, $C_t$ has CM order discriminant $\Delta=-252$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-11-1.
$t_Q([9,0,7])$:
1.858778600912632941396873411587150148096871799364498924593908322278514342304408597750229165520308004
comment: At $t=t_Q([9,0,7])$, $C_t$ has CM order discriminant $\Delta=-252$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-11-2.
$t_Q([9,-9,19])$:
2.856658782889076787843397190425790109555558313065475069778638003970787396923872121596842484907281724
comment: At $t=t_Q([9,-9,19])$, $C_t$ has CM order discriminant $\Delta=-603$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-13-1.
$t_Q([153,3,1])$:
0.9691978412080770865938565792352516111061244843138424001708301942974602248097061220375285157952776341
comment: At $t=t_Q([153,3,1])$, $C_t$ has CM order discriminant $\Delta=-603$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-13-2.
$t_Q([9,-9,43])$:
4.456569758290772218514613957084087433066276543110468674631599724209301776677274197440446475786742749
comment: At $t=t_Q([9,-9,43])$, $C_t$ has CM order discriminant $\Delta=-1467$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-14-1.
$t_Q([369,3,1])$:
0.9691978416583511537990214174258451895192084431620803448671455489456450213163986155470536245619055954
comment: At $t=t_Q([369,3,1])$, $C_t$ has CM order discriminant $\Delta=-1467$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-14-2.
$t_Q([4,0,1])$:
1.168548340725896444544528984583902898462863299940993950279068613355694130571781199439373184550840166
comment: At $t=t_Q([4,0,1])$, $C_t$ has CM order discriminant $\Delta=-16$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-16-1.
$t_Q([1,0,4])$:
4.188800666758732648823945163867100552661688085827012655883868478068625246659754841117933421134530334
comment: At $t=t_Q([1,0,4])$, $C_t$ has CM order discriminant $\Delta=-16$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-16-2.
$t_Q([7,0,1])$:
1.033189189715701859964114481289878222428981280515686891449013820888150612626002401645156475955465016
comment: At $t=t_Q([7,0,1])$, $C_t$ has CM order discriminant $\Delta=-28$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-17-1.
$t_Q([1,0,7])$:
5.541248768978044861890460337792477740664336880773014291760462805171381808533567427614052668959109125
comment: At $t=t_Q([1,0,7])$, $C_t$ has CM order discriminant $\Delta=-28$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-17-2.
$t_Q([3,-3,4])$:
2.175571576167239192802290507266762440359779818062497902521136719844114841628570224006537907132901618
comment: At $t=t_Q([3,-3,4])$, $C_t$ has CM order discriminant $\Delta=-39$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-18-1.
$t_Q([12,3,1])$:
0.9424972563215059630387981958387850056690868361258657015929408640786822523401876297495647782417056014
comment: At $t=t_Q([12,3,1])$, $C_t$ has CM order discriminant $\Delta=-39$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-18-2.
$t_Q([3,-3,19])$:
5.165702295527919319100605884556346898282062144516230164095796145527397917287011182647268509149330765
comment: At $t=t_Q([3,-3,19])$, $C_t$ has CM order discriminant $\Delta=-219$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-19-1.
$t_Q([57,3,1])$:
0.9691902261388042767299537199940102041212996784950773637922565314420397735774300929969626797724806948
comment: At $t=t_Q([57,3,1])$, $C_t$ has CM order discriminant $\Delta=-219$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-19-2.
$t_Q([3,-3,25])$:
5.954617287423112775828719163111304819592856598063640668330264406900615551494154513210771508153749758
comment: At $t=t_Q([3,-3,25])$, $C_t$ has CM order discriminant $\Delta=-291$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-20-1.
$t_Q([75,3,1])$:
0.9691970249345769694372203642843996831006052885951764839672566255150923665224406601191431598160904330
comment: At $t=t_Q([75,3,1])$, $C_t$ has CM order discriminant $\Delta=-291$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-20-2.
$t_Q([3,-3,31])$:
6.650597662369950064118422772945526618029897973454633889938550261933477797579218959472749520190063194
comment: At $t=t_Q([3,-3,31])$, $C_t$ has CM order discriminant $\Delta=-363$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-21-1.
$t_Q([93,3,1])$:
0.9691977292266128035065804265421594596579519785387796784367426248863023697682894740710884034556651310
comment: At $t=t_Q([93,3,1])$, $C_t$ has CM order discriminant $\Delta=-363$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-21-2.
$t_Q([3,-3,43])$:
7.859797244842470852432527574114519426963908937248622696276528257875164907764131566170407677321820842
comment: At $t=t_Q([3,-3,43])$, $C_t$ has CM order discriminant $\Delta=-507$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-22-1.
$t_Q([129,3,1])$:
0.9691978381524927922803758716803874783986114557280163388597568009146402173610085126178831456436334081
comment: At $t=t_Q([129,3,1])$, $C_t$ has CM order discriminant $\Delta=-507$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-22-2.
$t_Q([3,-3,61])$:
9.385912731444016061079931018878387348380662747830468064768803536812747805694980507295032070611826294
comment: At $t=t_Q([3,-3,61])$, $C_t$ has CM order discriminant $\Delta=-723$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-23-1.
$t_Q([183,3,1])$:
0.9691978416156319836501991214326394035142666428581032579722905250177877589291115455803725213860670244
comment: At $t=t_Q([183,3,1])$, $C_t$ has CM order discriminant $\Delta=-723$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-23-2.
$t_Q([3,-3,103])$:
12.22727400655258776040783460212488529802888926996721660612144006859902383495082903742129009967564549
comment: At $t=t_Q([3,-3,103])$, $C_t$ has CM order discriminant $\Delta=-1227$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-24-1.
$t_Q([309,3,1])$:
0.9691978416583405764316188105262693520220922013276985847489397915451921283499424683154464504726355271
comment: At $t=t_Q([309,3,1])$, $C_t$ has CM order discriminant $\Delta=-1227$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-24-2.
$t_Q([39,0,1])$:
0.9692707638765148467300861407854152880241959277266542885185461844859746720588577625131431844889667604
comment: At $t=t_Q([39,0,1])$, $C_t$ has CM order discriminant $\Delta=-156$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-25-1.
$t_Q([3,0,13])$:
4.359837336894815146704977178532255718677595636408737582590299861915821784761336548874938590801791033
comment: At $t=t_Q([3,0,13])$, $C_t$ has CM order discriminant $\Delta=-156$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [5], entry ν-N4-25-2.
Definition
The ordinary logarithmic Mahler measure [4] of the Hessian cubic $P_t=x^3+y^3+1-\sqrt[3]{t}xy$ [1] is $N(t)=\frac{1}{(2\pi)^2}\int_0^{2\pi}\int_0^{2\pi}\log|P_t(e^{i\theta},e^{i\phi})|\,d\theta\,d\phi$, using the real cube root for $t\in\mathbb R$.
Parameters
$t$
—   Hessian parameter ($t\in\mathbb R$)
Formulas
(1)
$N(t)=m\bigl(y^2+(x^2-kx)y+x\bigr)=\frac1\pi\int_0^\pi\log\max\{|y_1(e^{i\theta})|,|y_2(e^{i\theta})|\}\,d\theta,$ Here $k=\sqrt[3]{t}$ is real and $y_1(x),y_2(x)$ are the roots in $y$ of $y^2+(x^2-kx)y+x$. Their product is $x$, so on the unit circle the larger root has modulus at least one [1], Lemma 3.
(2)
$N(0)=\mu_3=\frac{3\sqrt3}{4\pi}L(2,\chi_{-3}),\qquad N(27)=3\mu_3,$ Here $\mu_3$ is the three-step Mahler measure and $\chi_{-3}(n)$ is $0,1,-1$ for $n\equiv0,1,2\pmod3$, respectively. The value $L(2,\chi_{-3})$ uses Conrey label $(3,2)$.
(3)
$\displaystyle\begin{aligned}C_t:\quad Y^2={}&X^3-27t^2X^2\\&+216t^3(t-27)X-432t^4(t-27)^2,\end{aligned}\qquad j(C_t)=\frac{t(t+216)^3}{(t-27)^3}. $ The model is nonsingular for $t\ne0,27$. If $t=t_Q([A,B,C])$, then $j(C_t)=j(\tau_Q([A,B,C]))$ and $H_\Delta(j(C_t))=0$, where $H_\Delta$ is the class polynomial of the imaginary quadratic order of discriminant $\Delta$ [2], Lemmas 2.3 and 2.4. The class-polynomial table contains examples for some of these orders, not necessarily every discriminant in the expanded catalogue.
(4)
$\displaystyle \tau_Q([A,B,C])=\frac{-B+i\sqrt{4AC-B^2}}{2A},\qquad t_Q(\tau)=27+\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{12}. $ The shorthand $t_Q([A,B,C])$ means $t_Q(\tau_Q([A,B,C]))$. Here $A,B,C\in\mathbb Z$, $A>0$, $\gcd(A,B,C)=1$ and $\Delta=B^2-4AC<0$. The Dedekind eta function is $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-e^{2\pi i n\tau})$ for $\operatorname{Im}\tau>0$. The level-three representatives follow [2], Sections 2 and 3.
Comments
(6)
Natural logarithms and normalized Haar measure are used. Zeros of $P_t$ on the torus are interpreted through the convergent logarithmic integral. In particular, for $-1<t<27$, $N(t)$ is the ordinary measure, which differs from the modified period introduced in [1].
(7)
The parameter is the cube of the coefficient $k$. Samart writes $n(k)$ [1], whereas Samart and Tao write $\mathrm n(t)$ [2]. The real cube root fixes the convention for negative $t$.
(8)
The CM arguments are all real parameters of degree at most four classified in [2], Section 3, and [5]. They include real embeddings of fields that also have nonreal embeddings, not only totally real fields. The quartic part contains 96 arguments in 24 totally real orbits and 44 real embeddings of non-totally-real fields; the four-dimensional determinants in [2], Theorem 5.4, concern only the totally real orbits. The degree bound applies to $t$, not $k=\sqrt[3]{t}$, and describes the present catalogue rather than the domain of $N$.
(9)
At $t=0$, the torus covering $(x,y)\mapsto(x^3,y^3)$ gives $N(0)=\mu_3$, the three-step Mahler measure. At $t=27$, the factorization of the cubic into three linear factors gives $N(27)=3\mu_3$.
(10)
In $t_Q([A,B,C])$, the triple specifies the CM point $\tau_Q([A,B,C])$ in (4), while $t_Q$ is the level-three modular function giving the Hessian argument. The order discriminant $\Delta=B^2-4AC$ belongs to an imaginary quadratic order; it is not the discriminant of the parameter field $\mathbb Q(t)$. Neither $\Delta$ nor the field degree alone identifies the argument or its embedding.
(11)
The additional integers $1\leq t\leq26$ are the arguments studied for the modified period in [1], Table 2; their ordinary measures provide companions for comparing the two normalizations, not the same proposed $L$-value identities. The exterior arguments $t=-27,-8,32,125$ come from the proved evaluations listed in [1], Table 1.
Programs
(P1)
Sage
from generate import Table

table = Table()
print(table.value({'t': '24'}, 100)['number'])
References
[1]
D. Samart, Mahler measure of a nonreciprocal family of elliptic curves, corrected version 3 (2023). (arXiv)
[2]
D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026). (arXiv)
[3]
Mathew Rogers and Wadim Zudilin, From L-series of elliptic curves to Mahler measures, version 3 (2011). (arXiv)
Links
Similar tables
Mahler_measures_of_(short_random_walks) —   holds $\mu_3=N(0)$, also giving the singular-fibre value $N(27)=3\mu_3$
Mahler_measures_of_x_x-1_y_y-1_k —   holds logarithmic Mahler measures of the reciprocal Laurent family $x+x^{-1}+y+y^{-1}+k$
Hilbert_class_polynomials —   holds class polynomials that identify CM orders; its present coverage need not include every discriminant in this expanded parameter catalogue
Values_of_Dirichlet_L-functions_at_positive_integers —   holds $L(2,\chi_{-3})$, the Dirichlet value in the evaluations at $t=0$ and $t=27$
Modified Mahler measures of the Hessian family —   holds the signed modified period $\widetilde N(t)=I(t)-2J(t)$ for $-1<t<27$, with the same parameter $t=k^3$ and real cube root; the ordinary measure here is $N(t)=I(t)+J(t)$, so $\widetilde N(t)=N(t)-3J(t)$
Quadratic CM determinants of Hessian Mahler measures —   forms $D(t)=a\nu(t)^2+b\nu(t^\sigma)^2$ at oriented quadratic CM arguments with fixed coefficients $a,b$; its scalar input $\nu(u)$ is the ordinary measure $N(u)$ from this table when $u\notin(-1,27)$, and the modified period when $-1<u<27$
Mahler measures of $(1+x)(1+y)(x+y)-kxy$ —   holds $g(4)=3N(32)/4$, a proved connection between the two families
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds 209 arguments: all 177 real CM parameters of degree at most four in [2] and [5], the intrinsic points $-1,0,27$, the integers $1\leq t\leq26$ studied for the modified period, and $-27,-8,32,125$ from [1], Table 1; overlaps are counted once. The CM selection contains 3 rational, 30 quadratic, 4 cubic and 140 quartic real arguments, including real embeddings of non-totally-real fields; no continuous mesh is used)
How they were obtained:

The attached generator evaluates the quadratic Jensen integral by adaptive tanh-sinh quadrature, split at the torus crossing and at geometrically spaced points around the narrow unit-modulus transition near $t=27$. A stable real expression avoids subtraction loss in the root-modulus integrand. For 100-digit output it uses nominal precisions of 140 and 190 digits with 40 extra integration digits (180 and 230 effective working digits), retaining 135 and 185 intermediate digits. The estimates and precision comparisons are heuristic, not certified error bounds.

more

All 209 entries were recomputed; nonzero values agree at the two nominal precisions to at least 127.66 relative decimal digits. The table retains 100 significant digits.

An independent implementation integrates the original cubic in the second variable at nominal precisions of 125 and 145 digits, using 40 extra arithmetic digits (165 and 185 working digits). Geometric subdivision resolves the narrow transition near $t=27$. It agrees with every ordinary value to over 110 relative digits. Independent Dirichlet and proved elliptic L-value controls are recorded in the attached verification evidence.

The CM arguments are reconstructed from integer minimal polynomials and exact rational isolating intervals. Exact class-polynomial relations and independent eta-product enclosures identify their level-three CM forms and embeddings. All 66 order discriminants were checked with Sage/Arb, PARI and direct ball products; 49 overlap the present class-polynomial table. These argument checks do not certify the numerical period integrals.

The source hashes, complete checks and preservation tests are in verification.json. Earlier attached numerical reports describe their original selections; argument-selection.md distinguishes them from the expanded revision.



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