Regulators of real quadratic fields

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Numbers

$D$

$R_K$

0.4812118250596034474977589134243684231351843343856605196610181688401638676082217744120094291227234750

comment: $\mathbb{Q}(\sqrt{5})$: $\varepsilon_K=(1+\sqrt{5})/2$, $N(\varepsilon_K)=-1$; $R_K=\log\varphi$, the logarithm of the golden ratio

0.8813735870195430252326093249797923090281603282616354107532956086533771842220260878337068919102560429

comment: $\mathbb{Q}(\sqrt{2})$: $\varepsilon_K=1+\sqrt{2}$, $N(\varepsilon_K)=-1$; $R_K=\log(1+\sqrt{2})=\operatorname{arsinh}(1)$

12:

1.316957896924816708625046347307968444026981971467516479768472256920460185416443976074219013450101784

comment: $\mathbb{Q}(\sqrt{3})$: $\varepsilon_K=2+\sqrt{3}$, $N(\varepsilon_K)=1$; $R_K=\log(2+\sqrt{3})=\operatorname{arcosh}(2)$

13:

1.194763217287109304111930828519090523536162075153005429270680299461324095830962530268871614289314375

comment: $\mathbb{Q}(\sqrt{13})$: $\varepsilon_K=(3+\sqrt{13})/2$, $N(\varepsilon_K)=-1$

17:

2.094712547261101294244822846065528653453151048198673265886805979926247949695627249150429365488740801

comment: $\mathbb{Q}(\sqrt{17})$: $\varepsilon_K=4+\sqrt{17}$, $N(\varepsilon_K)=-1$

21:

1.566799236972411078664056862580483493862082351092658863932945998012214813469392269627996849962220114

comment: $\mathbb{Q}(\sqrt{21})$: $\varepsilon_K=(5+\sqrt{21})/2$, $N(\varepsilon_K)=1$

24:

2.292431669561177687800787311348015431621868240015710247605016444831347855723856573484055076085019256

comment: $\mathbb{Q}(\sqrt{6})$: $\varepsilon_K=5+2\sqrt{6}$, $N(\varepsilon_K)=1$

28:

2.768659383313573832732001409383745519872057893445600633657277561381764216167126915163756411003609295

comment: $\mathbb{Q}(\sqrt{7})$: $\varepsilon_K=8+3\sqrt{7}$, $N(\varepsilon_K)=1$

29:

1.647231146371095710624858610443619663504414430193236528220310093084398375763310407877842025506942491

comment: $\mathbb{Q}(\sqrt{29})$: $\varepsilon_K=(5+\sqrt{29})/2$, $N(\varepsilon_K)=-1$

33:

3.828168471333101014963363758757313457551225484632143122563566897063799285244534035578125984357219626

comment: $\mathbb{Q}(\sqrt{33})$: $\varepsilon_K=23+4\sqrt{33}$, $N(\varepsilon_K)=1$

37:

2.491779852644911970429792537156622761621049578631067373170110652155832035515409458024208425946129102

comment: $\mathbb{Q}(\sqrt{37})$: $\varepsilon_K=6+\sqrt{37}$, $N(\varepsilon_K)=-1$

40:

1.818446459232066823483698963560708993786253942768121617451744167233054107866175751026084044360792694

comment: $\mathbb{Q}(\sqrt{10})$: $\varepsilon_K=3+\sqrt{10}$, $N(\varepsilon_K)=-1$

41:

4.159127134626180013108544973572201287085127627726096675004308327761140308125009213905572056956138833

comment: $\mathbb{Q}(\sqrt{41})$: $\varepsilon_K=32+5\sqrt{41}$, $N(\varepsilon_K)=-1$

44:

2.993222846126380897912667713774182913083660451180980642685145600977499226709739878280630962707130629

comment: $\mathbb{Q}(\sqrt{11})$: $\varepsilon_K=10+3\sqrt{11}$, $N(\varepsilon_K)=1$

53:

1.965720471649651521238756348670069302283699388407253100312302742379233233144246986458466187176810303

comment: $\mathbb{Q}(\sqrt{53})$: $\varepsilon_K=(7+\sqrt{53})/2$, $N(\varepsilon_K)=-1$

56:

3.400084414113339500700187214244867064519782873143950778863694555175049936039146224586818365308646527

comment: $\mathbb{Q}(\sqrt{14})$: $\varepsilon_K=15+4\sqrt{14}$, $N(\varepsilon_K)=1$

57:

5.710416052763152522543082381529571773891267742457577178777509198888377823787070885692485233446943978

comment: $\mathbb{Q}(\sqrt{57})$: $\varepsilon_K=151+20\sqrt{57}$, $N(\varepsilon_K)=1$

60:

2.063437068895560546727281172620131871456591449883392499836032692765902842847409911780353006403580326

comment: $\mathbb{Q}(\sqrt{15})$: $\varepsilon_K=4+\sqrt{15}$, $N(\varepsilon_K)=1$

61:

3.664218460886437525925846488464292734783877124104040903147749413270375780579198481768206724960690319

comment: $\mathbb{Q}(\sqrt{61})$: $\varepsilon_K=(39+5\sqrt{61})/2$, $N(\varepsilon_K)=-1$

65:

2.776472280723717673530804027028543458802828938983895985416131155030226809679611640402927505767753251

comment: $\mathbb{Q}(\sqrt{65})$: $\varepsilon_K=8+\sqrt{65}$, $N(\varepsilon_K)=-1$

69:

3.217271971157257876255315585381411341351206558884243566266455058728955257874538807863329768195266715

comment: $\mathbb{Q}(\sqrt{69})$: $\varepsilon_K=(25+3\sqrt{69})/2$, $N(\varepsilon_K)=1$

73:

7.666690419258287747402075701839906700456138252454970719058370945922684051649129760525019045417204075

comment: $\mathbb{Q}(\sqrt{73})$: $\varepsilon_K=1068+125\sqrt{73}$, $N(\varepsilon_K)=-1$

76:

5.828936966978926554734566857522909539067228954513068040492649192740640546795402817565243395082523177

comment: $\mathbb{Q}(\sqrt{19})$: $\varepsilon_K=170+39\sqrt{19}$, $N(\varepsilon_K)=1$

77:

2.184643791605108726676278133072127771464003355589154969885731658998881931911181279154284203272736586

comment: $\mathbb{Q}(\sqrt{77})$: $\varepsilon_K=(9+\sqrt{77})/2$, $N(\varepsilon_K)=1$

85:

2.209347708615334277705676480905806997047969581371916413904179186585501052658559764878948857162532583

comment: $\mathbb{Q}(\sqrt{85})$: $\varepsilon_K=(9+\sqrt{85})/2$, $N(\varepsilon_K)=-1$

88:

5.976344467430951637905983352354106112411086187969108044814913065134056996617160909793529117754412351

comment: $\mathbb{Q}(\sqrt{22})$: $\varepsilon_K=197+42\sqrt{22}$, $N(\varepsilon_K)=1$

89:

6.907756278980637055387298947411625879136882941272422819678047522269283366943623096444536361859282717

comment: $\mathbb{Q}(\sqrt{89})$: $\varepsilon_K=500+53\sqrt{89}$, $N(\varepsilon_K)=-1$

92:

3.870766700287093755598116798570745149594987411566645930676437300036643487600649072679185365785170319

comment: $\mathbb{Q}(\sqrt{23})$: $\varepsilon_K=24+5\sqrt{23}$, $N(\varepsilon_K)=1$

93:

3.366104642925109004985161631212215101744549079561427057892916909966100363351883348741256993128588435

comment: $\mathbb{Q}(\sqrt{93})$: $\varepsilon_K=(29+3\sqrt{93})/2$, $N(\varepsilon_K)=1$

97:

9.324383095977405318002401820904055233120308558854559104301190421399017230326076678454417435051703033

comment: $\mathbb{Q}(\sqrt{97})$: $\varepsilon_K=5604+569\sqrt{97}$, $N(\varepsilon_K)=-1$

101:

2.998222950297969738846595537596453476607058054877303655734459262753089657352166089224592755239128930

comment: $\mathbb{Q}(\sqrt{101})$: $\varepsilon_K=10+\sqrt{101}$, $N(\varepsilon_K)=-1$

104:

2.312438341272752620253562341364414383658245072646559237167228990099132554838238932415005774178954786

comment: $\mathbb{Q}(\sqrt{26})$: $\varepsilon_K=5+\sqrt{26}$, $N(\varepsilon_K)=-1$

105:

4.406570493076975547179869369240690701112218747087669629839024020824713335630583124913074228962154836

comment: $\mathbb{Q}(\sqrt{105})$: $\varepsilon_K=41+4\sqrt{105}$, $N(\varepsilon_K)=1$

109:

5.564535086760474369883774087601031893913626051769880952877973574625669997174658968401752970616162798

comment: $\mathbb{Q}(\sqrt{109})$: $\varepsilon_K=(261+25\sqrt{109})/2$, $N(\varepsilon_K)=-1$

113:

7.347300115903921570484538628091256496177626477878230666348191199287197996868813596648173050917933637

comment: $\mathbb{Q}(\sqrt{113})$: $\varepsilon_K=776+73\sqrt{113}$, $N(\varepsilon_K)=-1$

120:

3.088969904844603017916479853736295094533454441136435131225532448209215351305027698930521085626874865

comment: $\mathbb{Q}(\sqrt{30})$: $\varepsilon_K=11+2\sqrt{30}$, $N(\varepsilon_K)=1$

124:

8.019612686193878627627840611247972998488428459040891949549088839015192908645060774678374651374765916

comment: $\mathbb{Q}(\sqrt{31})$: $\varepsilon_K=1520+273\sqrt{31}$, $N(\varepsilon_K)=1$

129:

10.42554980734810676783170703546360841692649373270717686876682296108682547894128256200857510707250317

comment: $\mathbb{Q}(\sqrt{129})$: $\varepsilon_K=16855+1484\sqrt{129}$, $N(\varepsilon_K)=1$

133:

5.153258180413702109194980502016214080699960290961327195829534852369198346695530810052051523387409996

comment: $\mathbb{Q}(\sqrt{133})$: $\varepsilon_K=(173+15\sqrt{133})/2$, $N(\varepsilon_K)=1$

136:

4.248291097914388695301580778451465693729432092097502906134991118124723415856875107233744902897076672

comment: $\mathbb{Q}(\sqrt{34})$: $\varepsilon_K=35+6\sqrt{34}$, $N(\varepsilon_K)=1$

137:

8.157083867224171955798900063686620767698301641760345856315884237774397413895345318070408961460821372

comment: $\mathbb{Q}(\sqrt{137})$: $\varepsilon_K=1744+149\sqrt{137}$, $N(\varepsilon_K)=-1$

140:

2.477888730288475004813950745074505449456397657294732029574792739813029133732752430190279186124874890

comment: $\mathbb{Q}(\sqrt{35})$: $\varepsilon_K=6+\sqrt{35}$, $N(\varepsilon_K)=1$

141:

5.246996370178386296099100535834635423650406851468299704811965229972095660042095174053878595956160249

comment: $\mathbb{Q}(\sqrt{141})$: $\varepsilon_K=95+8\sqrt{141}$, $N(\varepsilon_K)=1$

145:

3.179785437699878826916971527769468831212360981024682282774541967535455943415871137592356165202493408

comment: $\mathbb{Q}(\sqrt{145})$: $\varepsilon_K=12+\sqrt{145}$, $N(\varepsilon_K)=-1$

149:

4.111142500863215824918025579618188520292524952433567528823711946050989349915288928153692769822270886

comment: $\mathbb{Q}(\sqrt{149})$: $\varepsilon_K=(61+5\sqrt{149})/2$, $N(\varepsilon_K)=-1$

152:

4.303882428113997112009880107993364511544915342649096975893993035795801843981912588283896887541541963

comment: $\mathbb{Q}(\sqrt{38})$: $\varepsilon_K=37+6\sqrt{38}$, $N(\varepsilon_K)=1$

156:

3.911622765214588466921623284655825585201700236431464276276452347477215959146043124855732271762944381

comment: $\mathbb{Q}(\sqrt{39})$: $\varepsilon_K=25+4\sqrt{39}$, $N(\varepsilon_K)=1$

157:

5.361314206462789661767712604661442400646049000231651638362162372927578156760631277232255487915227491

comment: $\mathbb{Q}(\sqrt{157})$: $\varepsilon_K=(213+17\sqrt{157})/2$, $N(\varepsilon_K)=-1$

161:

10.06688109764147059616615197218627295909118841016733420376210633124624605100632661436477973183537003

comment: $\mathbb{Q}(\sqrt{161})$: $\varepsilon_K=11775+928\sqrt{161}$, $N(\varepsilon_K)=1$

165:

2.558978977028612551445541826256834569885883572961505068977616653698958595018474766004332812718532241

comment: $\mathbb{Q}(\sqrt{165})$: $\varepsilon_K=(13+\sqrt{165})/2$, $N(\varepsilon_K)=1$

168:

3.256613954800052409316227360599978522930945919421498167520760924052920152140858759143910806900230562

comment: $\mathbb{Q}(\sqrt{42})$: $\varepsilon_K=13+2\sqrt{42}$, $N(\varepsilon_K)=1$

172:

8.848509280268372720291860480501493458485459116173604820950821155271995726776479022892561501947581646

comment: $\mathbb{Q}(\sqrt{43})$: $\varepsilon_K=3482+531\sqrt{43}$, $N(\varepsilon_K)=1$

173:

2.570814678095696836821251380510825341243524897957980075281332478856566607813435838828302264949345163

comment: $\mathbb{Q}(\sqrt{173})$: $\varepsilon_K=(13+\sqrt{173})/2$, $N(\varepsilon_K)=-1$

177:

11.73483625668438394962082332600680201103811367582947678354880803334443459936022115167739061704761130

comment: $\mathbb{Q}(\sqrt{177})$: $\varepsilon_K=62423+4692\sqrt{177}$, $N(\varepsilon_K)=1$

181:

7.173958906946717135957207060144636978630428515222985366200889681383642970466242840966191069535090188

comment: $\mathbb{Q}(\sqrt{181})$: $\varepsilon_K=(1305+97\sqrt{181})/2$, $N(\varepsilon_K)=-1$

184:

10.79281810240533976708449730930252307422737797256325862595924654293290399545663888558954833559658813

comment: $\mathbb{Q}(\sqrt{46})$: $\varepsilon_K=24335+3588\sqrt{46}$, $N(\varepsilon_K)=1$

185:

4.912708947095866367424208567423656139297779767295221667685681939676282237016442436064175027101905719

comment: $\mathbb{Q}(\sqrt{185})$: $\varepsilon_K=68+5\sqrt{185}$, $N(\varepsilon_K)=-1$

188:

4.564239666858496643729964866047530627501769953575393791699579934265144094790827694855746085963418791

comment: $\mathbb{Q}(\sqrt{47})$: $\varepsilon_K=48+7\sqrt{47}$, $N(\varepsilon_K)=1$

193:

15.07631652324125239560907349314557581074008430087532822362202423629295596589785481958243721466169172

comment: $\mathbb{Q}(\sqrt{193})$: $\varepsilon_K=1764132+126985\sqrt{193}$, $N(\varepsilon_K)=-1$

197:

3.333477586883992525024956054169842272356822588573213687642904356828872507440341808020860492804042918

comment: $\mathbb{Q}(\sqrt{197})$: $\varepsilon_K=14+\sqrt{197}$, $N(\varepsilon_K)=-1$

201:

13.84525380921252960390486447432133889852511910701416035503613090666053416303122126609054958161498970

comment: $\mathbb{Q}(\sqrt{201})$: $\varepsilon_K=515095+36332\sqrt{201}$, $N(\varepsilon_K)=1$

204:

4.605070170984757159450572551513062915547943483476541577357007701924407306089001223304546454200823161

comment: $\mathbb{Q}(\sqrt{51})$: $\varepsilon_K=50+7\sqrt{51}$, $N(\varepsilon_K)=1$

205:

3.760658843532550523290241575435163032943549151507341437604886237206352777701513125207443594476011741

comment: $\mathbb{Q}(\sqrt{205})$: $\varepsilon_K=(43+3\sqrt{205})/2$, $N(\varepsilon_K)=1$

209:

11.44145094519617190166390517102312639364399110954490800538231618879364527206100139368587757995459013

comment: $\mathbb{Q}(\sqrt{209})$: $\varepsilon_K=46551+3220\sqrt{209}$, $N(\varepsilon_K)=1$

213:

4.290271735838551110841582061477600699097575818081820466517946218404210732029685086965917931372130465

comment: $\mathbb{Q}(\sqrt{213})$: $\varepsilon_K=(73+5\sqrt{213})/2$, $N(\varepsilon_K)=1$

217:

15.85518761867588709059567836050082216516740348433459249915253568017444155830507734303488592806210580

comment: $\mathbb{Q}(\sqrt{217})$: $\varepsilon_K=3844063+260952\sqrt{217}$, $N(\varepsilon_K)=1$

220:

5.181751987126265554312435619961088788228430778366448176902180204865382411748228653423213880930222422

comment: $\mathbb{Q}(\sqrt{55})$: $\varepsilon_K=89+12\sqrt{55}$, $N(\varepsilon_K)=1$

221:

2.703575830931402317333949637054513979927629019692078281204469665496254753948895874388920460887739893

comment: $\mathbb{Q}(\sqrt{221})$: $\varepsilon_K=(15+\sqrt{221})/2$, $N(\varepsilon_K)=1$

229:

2.712465305184343974680879510606130069899358941540579114563147069065217524399785068248425020566737123

comment: $\mathbb{Q}(\sqrt{229})$: $\varepsilon_K=(15+\sqrt{229})/2$, $N(\varepsilon_K)=-1$

232:

5.288292537319899146825629699339937678549200498378900210386233461700790431026062575593428825873592039

comment: $\mathbb{Q}(\sqrt{58})$: $\varepsilon_K=99+13\sqrt{58}$, $N(\varepsilon_K)=-1$

233:

10.74315638622508419418098684760660866260019817651494211913412348030308370828429430091151885041670725

comment: $\mathbb{Q}(\sqrt{233})$: $\varepsilon_K=23156+1517\sqrt{233}$, $N(\varepsilon_K)=-1$

236:

6.966023297108484670495018686043348266197053038447832005236048594276451501368287499784362819915490299

comment: $\mathbb{Q}(\sqrt{59})$: $\varepsilon_K=530+69\sqrt{59}$, $N(\varepsilon_K)=1$

237:

4.343636716660797197142058731905556363956689572205736962080745843048241023612952942509041773329471655

comment: $\mathbb{Q}(\sqrt{237})$: $\varepsilon_K=(77+5\sqrt{237})/2$, $N(\varepsilon_K)=1$

241:

18.77149349074031224752730151751132466166369973795605613050099962363755770336808149510967999320353437

comment: $\mathbb{Q}(\sqrt{241})$: $\varepsilon_K=71011068+4574225\sqrt{241}$, $N(\varepsilon_K)=-1$

248:

4.836218912841156470897282140826484492647507300812044243201907872907045563807546642875753037272710429

comment: $\mathbb{Q}(\sqrt{62})$: $\varepsilon_K=63+8\sqrt{62}$, $N(\varepsilon_K)=1$

249:

16.65503512078619791431165547682612120366980986974886991696439755220728805143527816641950382983704092

comment: $\mathbb{Q}(\sqrt{249})$: $\varepsilon_K=8553815+542076\sqrt{249}$, $N(\varepsilon_K)=1$

253:

7.528868967901925182232998957068654976060727619614925305547318960952881224402795578424317758199463546

comment: $\mathbb{Q}(\sqrt{253})$: $\varepsilon_K=(1861+117\sqrt{253})/2$, $N(\varepsilon_K)=1$

257:

3.466711037884724756638546043397530710257479359515616494725470326620290117589007598568408753545654032

comment: $\mathbb{Q}(\sqrt{257})$: $\varepsilon_K=16+\sqrt{257}$, $N(\varepsilon_K)=-1$

264:

4.867475273605341639635986051265983697540796080642409087575754782323138364738906862090337528670532311

comment: $\mathbb{Q}(\sqrt{66})$: $\varepsilon_K=65+8\sqrt{66}$, $N(\varepsilon_K)=1$

265:

9.404590506416141582518377896491163528831428391138735024552804221912339685523754540326173367481594521

comment: $\mathbb{Q}(\sqrt{265})$: $\varepsilon_K=6072+373\sqrt{265}$, $N(\varepsilon_K)=-1$

268:

11.48949305788230826248841021456566273419095226406566808365329155173549992307423502378763020794768398

comment: $\mathbb{Q}(\sqrt{67})$: $\varepsilon_K=48842+5967\sqrt{67}$, $N(\varepsilon_K)=1$

269:

5.099903606000664539472127890741904271706066256580947482035029126965611583168124991165599010531179490

comment: $\mathbb{Q}(\sqrt{269})$: $\varepsilon_K=82+5\sqrt{269}$, $N(\varepsilon_K)=-1$

273:

7.282073185082187376847325362983834630472516760111520940243631329598579234738453711715679503020191765

comment: $\mathbb{Q}(\sqrt{273})$: $\varepsilon_K=727+44\sqrt{273}$, $N(\varepsilon_K)=1$

277:

7.868254411981305622604720362012165027323426527275234956730711551778659870863540651638448330101714246

comment: $\mathbb{Q}(\sqrt{277})$: $\varepsilon_K=(2613+157\sqrt{277})/2$, $N(\varepsilon_K)=-1$

280:

6.218596151477128082317554153685156886194168079229033294960439640737649493482624868241782045495445669

comment: $\mathbb{Q}(\sqrt{70})$: $\varepsilon_K=251+30\sqrt{70}$, $N(\varepsilon_K)=1$

281:

14.57025318305761211069003100052624962686407392507317549445431389628531522994540788298434295637563571

comment: $\mathbb{Q}(\sqrt{281})$: $\varepsilon_K=1063532+63445\sqrt{281}$, $N(\varepsilon_K)=-1$

284:

8.847934732685050766401070635025975240584469746635310395061323269967513345738894072263271023346805543

comment: $\mathbb{Q}(\sqrt{71})$: $\varepsilon_K=3480+413\sqrt{71}$, $N(\varepsilon_K)=1$

285:

2.829735037524390275366101086116373812923632011582734436443931880878057363331760963104039724775711615

comment: $\mathbb{Q}(\sqrt{285})$: $\varepsilon_K=(17+\sqrt{285})/2$, $N(\varepsilon_K)=1$

293:

2.836655728968925473211426695931220955288376498980652461903926574223729487159300083567085510844381064

comment: $\mathbb{Q}(\sqrt{293})$: $\varepsilon_K=(17+\sqrt{293})/2$, $N(\varepsilon_K)=-1$

296:

4.454482477060509481640558353805380971012906690000869087678973515824015468063667387015732938717530624

comment: $\mathbb{Q}(\sqrt{74})$: $\varepsilon_K=43+5\sqrt{74}$, $N(\varepsilon_K)=-1$

301:

10.03210061807112108575703397163602007486355058898733302990664084582660455034335070807181182201317045

comment: $\mathbb{Q}(\sqrt{301})$: $\varepsilon_K=(22745+1311\sqrt{301})/2$, $N(\varepsilon_K)=1$

305:

6.885508624537382722969555479745046921128452121748116840235199855806026833658422273816587691631027365

comment: $\mathbb{Q}(\sqrt{305})$: $\varepsilon_K=489+28\sqrt{305}$, $N(\varepsilon_K)=1$

309:

8.526152893498102357412598424025420705569497821925276209825608950579510348993785374538937666074255216

comment: $\mathbb{Q}(\sqrt{309})$: $\varepsilon_K=(5045+287\sqrt{309})/2$, $N(\varepsilon_K)=1$

312:

4.663350082584310353402377338741475950882466282314789424443292355701806216338408719947906873743533389

comment: $\mathbb{Q}(\sqrt{78})$: $\varepsilon_K=53+6\sqrt{78}$, $N(\varepsilon_K)=1$

313:

19.35176052080201401657022497602918867397538630170788186662089640156255984878779188026946246750225525

comment: $\mathbb{Q}(\sqrt{313})$: $\varepsilon_K=126862368+7170685\sqrt{313}$, $N(\varepsilon_K)=-1$

316:

5.075134750444809859787695118478698803836805379095706823007705600266212394086374547015455515150086306

comment: $\mathbb{Q}(\sqrt{79})$: $\varepsilon_K=80+9\sqrt{79}$, $N(\varepsilon_K)=1$

317:

4.488762592517530670595542624718771055011890832210672129856477984444177332277159220908504963482045508

comment: $\mathbb{Q}(\sqrt{317})$: $\varepsilon_K=(89+5\sqrt{317})/2$, $N(\varepsilon_K)=-1$

321:

6.063779800314906156484656764098988086514175556791706068138474525313572635808296346149033338428424919

comment: $\mathbb{Q}(\sqrt{321})$: $\varepsilon_K=215+12\sqrt{321}$, $N(\varepsilon_K)=1$

328:

2.893443985885871378072718306626369246055584762462533199231699336214345506621541393310021475512173007

comment: $\mathbb{Q}(\sqrt{82})$: $\varepsilon_K=9+\sqrt{82}$, $N(\varepsilon_K)=-1$

329:

15.37425078807237874850483356930269989296784606852964639912063904290963612791424341321171580907992532

comment: $\mathbb{Q}(\sqrt{329})$: $\varepsilon_K=2376415+131016\sqrt{329}$, $N(\varepsilon_K)=1$

332:

5.099829245500619335478113833945732102551318887107339446461762721584204122672002389353910550694957500

comment: $\mathbb{Q}(\sqrt{83})$: $\varepsilon_K=82+9\sqrt{83}$, $N(\varepsilon_K)=1$

337:

21.43211640733848912819430265914094754621747500736555412547645754757064473659715134942374588477916841

comment: $\mathbb{Q}(\sqrt{337})$: $\varepsilon_K=1015827336+55335641\sqrt{337}$, $N(\varepsilon_K)=-1$

341:

5.624004473050585914173504000722845810732028165990698999489044018290259126925769938385330091053307054

comment: $\mathbb{Q}(\sqrt{341})$: $\varepsilon_K=(277+15\sqrt{341})/2$, $N(\varepsilon_K)=1$

344:

9.943188917078510457695653973643540615438718181902982174021211104923953536653756425858789753694613160

comment: $\mathbb{Q}(\sqrt{86})$: $\varepsilon_K=10405+1122\sqrt{86}$, $N(\varepsilon_K)=1$

345:

9.512073262181494540109910759753793688662344188261573740654296770007664092186224267288865772131870621

comment: $\mathbb{Q}(\sqrt{345})$: $\varepsilon_K=6761+364\sqrt{345}$, $N(\varepsilon_K)=1$

348:

4.025032660551618149174452469154710050379577703636445805630699070674329711877303318866434500664472205

comment: $\mathbb{Q}(\sqrt{87})$: $\varepsilon_K=28+3\sqrt{87}$, $N(\varepsilon_K)=1$

349:

9.821192312756573422782572128907937483179083093216657942990381983900297157666003716438448622787655914

comment: $\mathbb{Q}(\sqrt{349})$: $\varepsilon_K=9210+493\sqrt{349}$, $N(\varepsilon_K)=-1$

353:

11.86729375066626605563998726836226348692681740919174891420247998278461906525413652012357062073957527

comment: $\mathbb{Q}(\sqrt{353})$: $\varepsilon_K=71264+3793\sqrt{353}$, $N(\varepsilon_K)=-1$

357:

2.941657314651186076128172773975618199652699580731431099585672352375063131549671343993710567093901556

comment: $\mathbb{Q}(\sqrt{357})$: $\varepsilon_K=(19+\sqrt{357})/2$, $N(\varepsilon_K)=1$

364:

8.054522508628127484343262687989025597057797278607381987535130784527189441304388082097236277335040944

comment: $\mathbb{Q}(\sqrt{91})$: $\varepsilon_K=1574+165\sqrt{91}$, $N(\varepsilon_K)=1$

365:

2.947197622570033060867383142183555661593265380990876360863425970168643874660383991911419500422853671

comment: $\mathbb{Q}(\sqrt{365})$: $\varepsilon_K=(19+\sqrt{365})/2$, $N(\varepsilon_K)=-1$

373:

9.233666206823877404050775711434965565981804665963564398513570493317853130073531183535033242944972311

comment: $\mathbb{Q}(\sqrt{373})$: $\varepsilon_K=5118+265\sqrt{373}$, $N(\varepsilon_K)=-1$

376:

15.27100210303118287693252299751764957353073478303223811426236887700062883203093712148683571813553548

comment: $\mathbb{Q}(\sqrt{94})$: $\varepsilon_K=2143295+221064\sqrt{94}$, $N(\varepsilon_K)=1$

377:

6.144181029109401574125158212272009742503257568252906919816831051939694643656156759454716514788884506

comment: $\mathbb{Q}(\sqrt{377})$: $\varepsilon_K=233+12\sqrt{377}$, $N(\varepsilon_K)=1$

380:

4.356544420601752192181187278153033857496104548676724484026332086089165094253085293137557464256708645

comment: $\mathbb{Q}(\sqrt{95})$: $\varepsilon_K=39+4\sqrt{95}$, $N(\varepsilon_K)=1$

381:

7.615790829370307524702347458154450915559358728279551023189363641410300936435320970866422920494992554

comment: $\mathbb{Q}(\sqrt{381})$: $\varepsilon_K=1015+52\sqrt{381}$, $N(\varepsilon_K)=1$

385:

12.16348868296165521758668329393836769893969268972683797290855394153036559523347188009960055829246576

comment: $\mathbb{Q}(\sqrt{385})$: $\varepsilon_K=95831+4884\sqrt{385}$, $N(\varepsilon_K)=1$

389:

7.849323970152694681630728614204876518706443577189078113159625583961618600777612291757761043296900965

comment: $\mathbb{Q}(\sqrt{389})$: $\varepsilon_K=1282+65\sqrt{389}$, $N(\varepsilon_K)=-1$

393:

18.34675737322029098890415394849669945972943989300609467892779312600831298310679497401289699394236651

comment: $\mathbb{Q}(\sqrt{393})$: $\varepsilon_K=46437143+2342444\sqrt{393}$, $N(\varepsilon_K)=1$

397:

8.145259650679123304771912739902738205264195653530948711524203210918901657158468844134788818429586290

comment: $\mathbb{Q}(\sqrt{397})$: $\varepsilon_K=(3447+173\sqrt{397})/2$, $N(\varepsilon_K)=-1$

401:

3.689503868988905640821653570961093329678760832868086366520509251274995424465766662121747017302471762

comment: $\mathbb{Q}(\sqrt{401})$: $\varepsilon_K=20+\sqrt{401}$, $N(\varepsilon_K)=-1$

408:

5.308243189099001446313874981895365374706616841876010182426519701311521069018949098378880431180280704

comment: $\mathbb{Q}(\sqrt{102})$: $\varepsilon_K=101+10\sqrt{102}$, $N(\varepsilon_K)=1$

409:

26.13421340356027852009543152874913497238465518829717946451275734176507968535053581024599220176886813

comment: $\mathbb{Q}(\sqrt{409})$: $\varepsilon_K=111921796968+5534176685\sqrt{409}$, $N(\varepsilon_K)=-1$

412:

13.02817576727799214427222316184521798123139242169673965414636403744881086577057584650887889711945270

comment: $\mathbb{Q}(\sqrt{103})$: $\varepsilon_K=227528+22419\sqrt{103}$, $N(\varepsilon_K)=1$

413:

4.110605010811753147295121616363358802946930700896743837856228393742203021818950789290515677938672168

comment: $\mathbb{Q}(\sqrt{413})$: $\varepsilon_K=(61+3\sqrt{413})/2$, $N(\varepsilon_K)=1$

417:

18.95509765602493230057740928736958075624841266969077407702419017154722897037612453295526050254143393

comment: $\mathbb{Q}(\sqrt{417})$: $\varepsilon_K=85322647+4178268\sqrt{417}$, $N(\varepsilon_K)=1$

421:

13.00569247310560606694639825649815634123433340066616824945244904536232187513124562509213688480886930

comment: $\mathbb{Q}(\sqrt{421})$: $\varepsilon_K=(444939+21685\sqrt{421})/2$, $N(\varepsilon_K)=-1$

424:

8.988446055648415161048356183014684523733941578243964379315364755870607860485700138269879614444011460

comment: $\mathbb{Q}(\sqrt{106})$: $\varepsilon_K=4005+389\sqrt{106}$, $N(\varepsilon_K)=-1$

428:

7.562161361084939554337007217338817917851741810712150542557274883614178642451115572586765832881551268

comment: $\mathbb{Q}(\sqrt{107})$: $\varepsilon_K=962+93\sqrt{107}$, $N(\varepsilon_K)=1$

429:

4.976686176601255491838433086972897845859336270550700947237336932476066982051821410165501012515109403

comment: $\mathbb{Q}(\sqrt{429})$: $\varepsilon_K=(145+7\sqrt{429})/2$, $N(\varepsilon_K)=1$

433:

23.39474343041491157756845078794499265485674520727207943450975640692026170163131991933249296429271140

comment: $\mathbb{Q}(\sqrt{433})$: $\varepsilon_K=7230660684+347483377\sqrt{433}$, $N(\varepsilon_K)=-1$

437:

3.042247112093328531406186994901353275778576487866277754922463947563181017072818466366242437370133196

comment: $\mathbb{Q}(\sqrt{437})$: $\varepsilon_K=(21+\sqrt{437})/2$, $N(\varepsilon_K)=1$

440:

3.737102242198923900976805054113159386884204741143073889422198983717266311856623624779809037047570493

comment: $\mathbb{Q}(\sqrt{110})$: $\varepsilon_K=21+2\sqrt{110}$, $N(\varepsilon_K)=1$

444:

6.380119664149667076353327010542355585669659216369105898146610850145638407459257483396489686963178972

comment: $\mathbb{Q}(\sqrt{111})$: $\varepsilon_K=295+28\sqrt{111}$, $N(\varepsilon_K)=1$

445:

3.046782337219411019264568660830402309672843614753597989647786996836786774723599057537364443628154161

comment: $\mathbb{Q}(\sqrt{445})$: $\varepsilon_K=(21+\sqrt{445})/2$, $N(\varepsilon_K)=-1$

449:

19.75289546928047603249361784900711545299815559781462933498475919389302951168423154429581195455317930

comment: $\mathbb{Q}(\sqrt{449})$: $\varepsilon_K=189471332+8941705\sqrt{449}$, $N(\varepsilon_K)=-1$

453:

5.003901259885764179897729043115237824218930370528355308473930389072658989489688675846584124928926169

comment: $\mathbb{Q}(\sqrt{453})$: $\varepsilon_K=(149+7\sqrt{453})/2$, $N(\varepsilon_K)=1$

456:

7.625594834178769881382049053520343275908887643247071548669782668761042649135954527744222478379459209

comment: $\mathbb{Q}(\sqrt{114})$: $\varepsilon_K=1025+96\sqrt{114}$, $N(\varepsilon_K)=1$

457:

25.49547390350330616517989506045051342018143439554802784326921244034395497926075206169890453496545746

comment: $\mathbb{Q}(\sqrt{457})$: $\varepsilon_K=59089951584+2764111349\sqrt{457}$, $N(\varepsilon_K)=-1$

460:

7.719573792079356797053065089867662363004642677189926670439252718016570699660517751458591057189063427

comment: $\mathbb{Q}(\sqrt{115})$: $\varepsilon_K=1126+105\sqrt{115}$, $N(\varepsilon_K)=1$

461:

5.899904859596685834663496389955396656348604572169800475653610174641631164302837594177179222425407243

comment: $\mathbb{Q}(\sqrt{461})$: $\varepsilon_K=(365+17\sqrt{461})/2$, $N(\varepsilon_K)=-1$

465:

10.36539600307500178525551303267019329549388407676893378394889993325039757921401145812575272101555762

comment: $\mathbb{Q}(\sqrt{465})$: $\varepsilon_K=15871+736\sqrt{465}$, $N(\varepsilon_K)=1$

469:

4.174150499430208300828293391937026147495370220483139001674454945931851980655267021101285022087698592

comment: $\mathbb{Q}(\sqrt{469})$: $\varepsilon_K=(65+3\sqrt{469})/2$, $N(\varepsilon_K)=1$

472:

13.32747981226704125326159725932007220364443529775752947687602631802483981837930547771261931307427398

comment: $\mathbb{Q}(\sqrt{118})$: $\varepsilon_K=306917+28254\sqrt{118}$, $N(\varepsilon_K)=1$

473:

5.159022268115710484580241235863570970831917681362925482049020605258506923113557581448255670055243610

comment: $\mathbb{Q}(\sqrt{473})$: $\varepsilon_K=87+4\sqrt{473}$, $N(\varepsilon_K)=1$

476:

5.480621561778750480637401284604855252583346613337922884518505319160445135781429760705115060114715007

comment: $\mathbb{Q}(\sqrt{119})$: $\varepsilon_K=120+11\sqrt{119}$, $N(\varepsilon_K)=1$

481:

14.47213897182406812875851312685309372326763208388319440912334313064349747097182057629170651983418902

comment: $\mathbb{Q}(\sqrt{481})$: $\varepsilon_K=964140+43961\sqrt{481}$, $N(\varepsilon_K)=-1$

485:

3.784705763099432625451986840419791297950362702099210886358429593205833813744867052025586377859993761

comment: $\mathbb{Q}(\sqrt{485})$: $\varepsilon_K=22+\sqrt{485}$, $N(\varepsilon_K)=-1$

488:

3.093102195050827084219943190020506794357893619067853814889722583312170084508710910775436332414811721

comment: $\mathbb{Q}(\sqrt{122})$: $\varepsilon_K=11+\sqrt{122}$, $N(\varepsilon_K)=-1$

489:

23.44359105416490036924056470188718101533694498751099430476411527077589672033069270897051206346563368

comment: $\mathbb{Q}(\sqrt{489})$: $\varepsilon_K=7592629975+343350596\sqrt{489}$, $N(\varepsilon_K)=1$

492:

5.497151428309934927365135478315878921895365920091174526679834540518773122296150448228814192464114132

comment: $\mathbb{Q}(\sqrt{123})$: $\varepsilon_K=122+11\sqrt{123}$, $N(\varepsilon_K)=1$

493:

4.709611353676475685598244301090444603590414433548408771322501566850467700725724335454369523922703615

comment: $\mathbb{Q}(\sqrt{493})$: $\varepsilon_K=(111+5\sqrt{493})/2$, $N(\varepsilon_K)=-1$

497:

14.69255056023448566402229085376830274926648672486975307560653169468168364864621283231538197653721984

comment: $\mathbb{Q}(\sqrt{497})$: $\varepsilon_K=1201887+53912\sqrt{497}$, $N(\varepsilon_K)=1$

501:

10.24796338776260736276163756016788179887103438201043574157751141191696137579550585184457437585747525

comment: $\mathbb{Q}(\sqrt{501})$: $\varepsilon_K=(28225+1261\sqrt{501})/2$, $N(\varepsilon_K)=1$

505:

7.388945715636159066576769892981648077448676378504688453047183623362498249556679013798023544950230096

comment: $\mathbb{Q}(\sqrt{505})$: $\varepsilon_K=809+36\sqrt{505}$, $N(\varepsilon_K)=1$

508:

16.06271485621692213686131861845183461980278706473696368311058984555647501935397023702452753861386775

comment: $\mathbb{Q}(\sqrt{127})$: $\varepsilon_K=4730624+419775\sqrt{127}$, $N(\varepsilon_K)=1$

509:

6.829794906246680153508053917993746477079998199278696481415947970542773930306907906395034872815052779

comment: $\mathbb{Q}(\sqrt{509})$: $\varepsilon_K=(925+41\sqrt{509})/2$, $N(\varepsilon_K)=-1$

517:

9.266058851787048739941113783226038516562386063694689319784420264618233731561535143329313480369130438

comment: $\mathbb{Q}(\sqrt{517})$: $\varepsilon_K=(10573+465\sqrt{517})/2$, $N(\varepsilon_K)=1$

520:

4.736275386267656645278399135549461746657556413832299707271828508067982553913844709548076066199141153

comment: $\mathbb{Q}(\sqrt{130})$: $\varepsilon_K=57+5\sqrt{130}$, $N(\varepsilon_K)=-1$

521:

19.36363085550094548115001685213232850194256892011266451739784416987539225277050980602578023935262297

comment: $\mathbb{Q}(\sqrt{521})$: $\varepsilon_K=128377240+5624309\sqrt{521}$, $N(\varepsilon_K)=-1$

524:

9.962699409947175184240039737944953679927003703656047663966608148133630324387587193508390145565547314

comment: $\mathbb{Q}(\sqrt{131})$: $\varepsilon_K=10610+927\sqrt{131}$, $N(\varepsilon_K)=1$

533:

3.137379237316647317914866557085347330988064252129927208871749223819067953334526963025370246811581842

comment: $\mathbb{Q}(\sqrt{533})$: $\varepsilon_K=(23+\sqrt{533})/2$, $N(\varepsilon_K)=-1$

536:

12.58399525062019674980557646237924162666739526817152848603959190234288250411632987049531653781235922

comment: $\mathbb{Q}(\sqrt{134})$: $\varepsilon_K=145925+12606\sqrt{134}$, $N(\varepsilon_K)=1$

537:

19.76797157593281937206588991338753393410626395986618021589147263952178871554467660876233361627211608

comment: $\mathbb{Q}(\sqrt{537})$: $\varepsilon_K=192349463+8300492\sqrt{537}$, $N(\varepsilon_K)=1$

541:

14.14942595723287553057144982437190520131843944793266877921772751868269082166386678420743354048020217

comment: $\mathbb{Q}(\sqrt{541})$: $\varepsilon_K=(1396425+60037\sqrt{541})/2$, $N(\varepsilon_K)=-1$

545:

8.274356941745588388281467221223466110768410340073948373150935735319701933733467531277110907290015635

comment: $\mathbb{Q}(\sqrt{545})$: $\varepsilon_K=1961+84\sqrt{545}$, $N(\varepsilon_K)=1$

552:

4.543181589671228526574948115782692396528299038703776979784533376267898619811157017888188415738771318

comment: $\mathbb{Q}(\sqrt{138})$: $\varepsilon_K=47+4\sqrt{138}$, $N(\varepsilon_K)=1$

553:

27.85358183728156831890546596272323068456384550524569618685606817355985583276611875012870220503277613

comment: $\mathbb{Q}(\sqrt{553})$: $\varepsilon_K=624635837407+26562217704\sqrt{553}$, $N(\varepsilon_K)=1$

556:

18.85975147106356802403292644775515769994341002693190620722194330018516529133433629112167699122807874

comment: $\mathbb{Q}(\sqrt{139})$: $\varepsilon_K=77563250+6578829\sqrt{139}$, $N(\varepsilon_K)=1$

557:

5.463849759152821330464871919977989268445044779126150160907893081797281521801024045600682152904422040

comment: $\mathbb{Q}(\sqrt{557})$: $\varepsilon_K=118+5\sqrt{557}$, $N(\varepsilon_K)=-1$

561:

13.86007274921933021155913263012504371221612239023849409082181493477725648179965666579029858272929187

comment: $\mathbb{Q}(\sqrt{561})$: $\varepsilon_K=522785+22072\sqrt{561}$, $N(\varepsilon_K)=1$

565:

5.733351750021093916867739372885546264541736077813065213204136319009457721626394375157767819434389102

comment: $\mathbb{Q}(\sqrt{565})$: $\varepsilon_K=(309+13\sqrt{565})/2$, $N(\varepsilon_K)=-1$

568:

5.655979585058949815802003002904922200337812999536070430308604712612105092542645443636119863101117977

comment: $\mathbb{Q}(\sqrt{142})$: $\varepsilon_K=143+12\sqrt{142}$, $N(\varepsilon_K)=1$

569:

22.47935109184652375239598254614983146902121272328907085345960511147984709070501866177965233913900132

comment: $\mathbb{Q}(\sqrt{569})$: $\varepsilon_K=2894863832+121359005\sqrt{569}$, $N(\varepsilon_K)=-1$

572:

3.176313180591655766889219413387377315148600736394506676486909403417508062820906101717742738833573358

comment: $\mathbb{Q}(\sqrt{143})$: $\varepsilon_K=12+\sqrt{143}$, $N(\varepsilon_K)=1$

573:

6.641180465450296119489794636402097974446999275381404666385880543931171016274969773993587123659208733

comment: $\mathbb{Q}(\sqrt{573})$: $\varepsilon_K=383+16\sqrt{573}$, $N(\varepsilon_K)=1$

577:

3.871634756387731429687905076632625809227717018978660971040385654301256987603772728334184167563777749

comment: $\mathbb{Q}(\sqrt{577})$: $\varepsilon_K=24+\sqrt{577}$, $N(\varepsilon_K)=-1$

581:

8.813587182358680793671874748415886032980328031399115474869027326406219451291858417025200144979976173

comment: $\mathbb{Q}(\sqrt{581})$: $\varepsilon_K=(6725+279\sqrt{581})/2$, $N(\varepsilon_K)=1$

584:

5.669869032162013398068023536983466451673303465562703271569641924447512036661519099294396158953052079

comment: $\mathbb{Q}(\sqrt{146})$: $\varepsilon_K=145+12\sqrt{146}$, $N(\varepsilon_K)=1$

589:

15.28783971520717202331954604640268854851983179247464767477798574531431057346223308146323553422049532

comment: $\mathbb{Q}(\sqrt{589})$: $\varepsilon_K=(4359377+179625\sqrt{589})/2$, $N(\varepsilon_K)=1$

593:

13.99888489372595367627946529982241205737326317991672983719177444674053415618680858375334946349983945

comment: $\mathbb{Q}(\sqrt{593})$: $\varepsilon_K=600632+24665\sqrt{593}$, $N(\varepsilon_K)=-1$

597:

9.184919984107718068111234133828835900399266619615134390322307208853733831275824581263155071137870865

comment: $\mathbb{Q}(\sqrt{597})$: $\varepsilon_K=(9749+399\sqrt{597})/2$, $N(\varepsilon_K)=1$

601:

33.26200565816615950759892611771094864213644436408927343415056358004103381068964980247133831737192348

comment: $\mathbb{Q}(\sqrt{601})$: $\varepsilon_K=139468303679532+5689030769845\sqrt{601}$, $N(\varepsilon_K)=-1$

604:

21.96346335551494071927079940487510402493925977998030022282137160671603770240910031274589110390815688

comment: $\mathbb{Q}(\sqrt{151})$: $\varepsilon_K=1728148040+140634693\sqrt{151}$, $N(\varepsilon_K)=1$

609:

14.00727901858255800193258720996268968051689830940554924329068969496525693342805368534672718649376649

comment: $\mathbb{Q}(\sqrt{609})$: $\varepsilon_K=605695+24544\sqrt{609}$, $N(\varepsilon_K)=1$

613:

11.50047831977099165079561991015211299268075232831274427271006243654303161542488949846499263498778509

comment: $\mathbb{Q}(\sqrt{613})$: $\varepsilon_K=(98763+3989\sqrt{613})/2$, $N(\varepsilon_K)=-1$

616:

10.65937476236596332550807051757364567851865500878892934096030497423947881697638022716986846231938329

comment: $\mathbb{Q}(\sqrt{154})$: $\varepsilon_K=21295+1716\sqrt{154}$, $N(\varepsilon_K)=1$

617:

18.22246675276399884686664228698858017278392047470921969355188973871946992163400438885905720270959205

comment: $\mathbb{Q}(\sqrt{617})$: $\varepsilon_K=41009716+1650989\sqrt{617}$, $N(\varepsilon_K)=-1$

620:

6.210596044807235690132212491629744159902687586775708829052259758745126145248322930952137209002565382

comment: $\mathbb{Q}(\sqrt{155})$: $\varepsilon_K=249+20\sqrt{155}$, $N(\varepsilon_K)=1$

629:

3.220471998464453038387360429671804784138735629772886448019043183401911879441924387984825352357741850

comment: $\mathbb{Q}(\sqrt{629})$: $\varepsilon_K=(25+\sqrt{629})/2$, $N(\varepsilon_K)=-1$

632:

9.647691664776808686166306282302710065237124545322189324484258538720773149685891739555547913617367829

comment: $\mathbb{Q}(\sqrt{158})$: $\varepsilon_K=7743+616\sqrt{158}$, $N(\varepsilon_K)=1$

633:

20.59718603385451122085442138320546475597228419704991131013181847920431136566132913601595253541126934

comment: $\mathbb{Q}(\sqrt{633})$: $\varepsilon_K=440772247+17519124\sqrt{633}$, $N(\varepsilon_K)=1$

636:

7.881559774442251805632603033928400297339811515847366032339948731925510358462744278004637396096548416

comment: $\mathbb{Q}(\sqrt{159})$: $\varepsilon_K=1324+105\sqrt{159}$, $N(\varepsilon_K)=1$

641:

25.00328282076123256527742535309970830823723934910135541055433535046811899596515527481924040465392340

comment: $\mathbb{Q}(\sqrt{641})$: $\varepsilon_K=36120833468+1426687145\sqrt{641}$, $N(\varepsilon_K)=-1$

645:

4.844125080567773400227441387782500816479440272416108265196656482467449861827504761167102395640038164

comment: $\mathbb{Q}(\sqrt{645})$: $\varepsilon_K=(127+5\sqrt{645})/2$, $N(\varepsilon_K)=1$

649:

35.34845536300888136304222747531205123386885724180640754625118973573402752905611207718545011816368032

comment: $\mathbb{Q}(\sqrt{649})$: $\varepsilon_K=1123593226162199+44104892095380\sqrt{649}$, $N(\varepsilon_K)=1$

652:

18.66879044702706229352862448269684010429426388267751937024121306238642635362443834213636688523469127

comment: $\mathbb{Q}(\sqrt{163})$: $\varepsilon_K=64080026+5019135\sqrt{163}$, $N(\varepsilon_K)=1$

653:

7.415175472073639445836335517085753833210761603451536469090407906498886756842083527250711602363342361

comment: $\mathbb{Q}(\sqrt{653})$: $\varepsilon_K=(1661+65\sqrt{653})/2$, $N(\varepsilon_K)=-1$

661:

14.39746860274819850330520447460632970155988999498122827929041110022657564234630182985377626877207155

comment: $\mathbb{Q}(\sqrt{661})$: $\varepsilon_K=(1789539+69605\sqrt{661})/2$, $N(\varepsilon_K)=-1$

664:

21.94757204826829150393307604072265605733708656019210609812312928937974936352003405991518292071294703

comment: $\mathbb{Q}(\sqrt{166})$: $\varepsilon_K=1700902565+132015642\sqrt{166}$, $N(\varepsilon_K)=1$

665:

10.21968419155781347613499180296035752381172012192783628835311098673241262606719110909672729410367106

comment: $\mathbb{Q}(\sqrt{665})$: $\varepsilon_K=13719+532\sqrt{665}$, $N(\varepsilon_K)=1$

668:

5.817102302135762817626824543353097962763007606618735592068530968819962284652059162169726521380535522

comment: $\mathbb{Q}(\sqrt{167})$: $\varepsilon_K=168+13\sqrt{167}$, $N(\varepsilon_K)=1$

669:

12.62900104550391297740689690332006777871386115464962624722633558488134133387780850722803972951382296

comment: $\mathbb{Q}(\sqrt{669})$: $\varepsilon_K=(305285+11803\sqrt{669})/2$, $N(\varepsilon_K)=1$

673:

32.21217429457924065080747280824518701157528675252171258839141515845678156863266638264771402848312231

comment: $\mathbb{Q}(\sqrt{673})$: $\varepsilon_K=48813455293932+1881620424025\sqrt{673}$, $N(\varepsilon_K)=-1$

677:

3.951613336082065540026529081051288940142624612031944998761102519450424054583103508329808778883341386

comment: $\mathbb{Q}(\sqrt{677})$: $\varepsilon_K=26+\sqrt{677}$, $N(\varepsilon_K)=-1$

680:

3.259572556262921561296584731338444312702859086841239058030125361888205720028783232423847117075310066

comment: $\mathbb{Q}(\sqrt{170})$: $\varepsilon_K=13+\sqrt{170}$, $N(\varepsilon_K)=-1$

681:

30.69843812829357752335081873923163529026214218906903484943287287823880876394523110033086206912290774

comment: $\mathbb{Q}(\sqrt{681})$: $\varepsilon_K=10743166003415+411679015748\sqrt{681}$, $N(\varepsilon_K)=1$

685:

6.632003513258114905870712376106410254374701829408792532039434086248889482835281089109374614853523807

comment: $\mathbb{Q}(\sqrt{685})$: $\varepsilon_K=(759+29\sqrt{685})/2$, $N(\varepsilon_K)=-1$

689:

5.347084854209184791455226172907609736460518412297642569288446221152108184780999649242605763777500486

comment: $\mathbb{Q}(\sqrt{689})$: $\varepsilon_K=105+4\sqrt{689}$, $N(\varepsilon_K)=1$

696:

7.973155314701886653592549633557043551355476011795030014233559950632958775124077117670034840159613284

comment: $\mathbb{Q}(\sqrt{174})$: $\varepsilon_K=1451+110\sqrt{174}$, $N(\varepsilon_K)=1$

697:

5.575963450863238908574232161780591650241918120432123645320308953741229198354286349568902537721208889

comment: $\mathbb{Q}(\sqrt{697})$: $\varepsilon_K=132+5\sqrt{697}$, $N(\varepsilon_K)=-1$

701:

10.06747540444252926422595184726285967154966937776589259828293389646529342750995445283232309143177223

comment: $\mathbb{Q}(\sqrt{701})$: $\varepsilon_K=11782+445\sqrt{701}$, $N(\varepsilon_K)=-1$

705:

13.06964169493057294368951569443208839330030130967211401971757274388647879123067782386856995189925757

comment: $\mathbb{Q}(\sqrt{705})$: $\varepsilon_K=237161+8932\sqrt{705}$, $N(\varepsilon_K)=1$

709:

17.41256579922471288234053070442174250948348284458904200385830196674801105444704474565044911101641629

comment: $\mathbb{Q}(\sqrt{709})$: $\varepsilon_K=18245310+685217\sqrt{709}$, $N(\varepsilon_K)=-1$

712:

8.071530796022351693010529459497991521676232884549810036719466244095231177327588159031634731421858425

comment: $\mathbb{Q}(\sqrt{178})$: $\varepsilon_K=1601+120\sqrt{178}$, $N(\varepsilon_K)=1$

713:

16.17378898097577954103416037048804333071156210546858848214510191480368548325496687848444023443668761

comment: $\mathbb{Q}(\sqrt{713})$: $\varepsilon_K=5286367+197976\sqrt{713}$, $N(\varepsilon_K)=1$

716:

15.94140859053406203830813690476470894490791420451687684364885064511682044242207409109232827042817043

comment: $\mathbb{Q}(\sqrt{179})$: $\varepsilon_K=4190210+313191\sqrt{179}$, $N(\varepsilon_K)=1$

717:

5.484779715711570789183727900773931788298882875774810524771619389643481451864068703824617749227827515

comment: $\mathbb{Q}(\sqrt{717})$: $\varepsilon_K=(241+9\sqrt{717})/2$, $N(\varepsilon_K)=1$

721:

38.15681360474933359851872903834760016348452117804608012220313333093961531822280333637172958169889523

comment: $\mathbb{Q}(\sqrt{721})$: $\varepsilon_K=18632176943292415+693898530122112\sqrt{721}$, $N(\varepsilon_K)=1$

728:

3.988640934494431672198188691964255174293007716616619000402283190457115765502581401492285892430938937

comment: $\mathbb{Q}(\sqrt{182})$: $\varepsilon_K=27+2\sqrt{182}$, $N(\varepsilon_K)=1$

732:

6.881410249540206007077047133922061539908013227512702666872674593206510624471173739488228541282380140

comment: $\mathbb{Q}(\sqrt{183})$: $\varepsilon_K=487+36\sqrt{183}$, $N(\varepsilon_K)=1$

733:

3.297205794175233866494236848337215013455668916260815648405295207221328467042097081898869761443826805

comment: $\mathbb{Q}(\sqrt{733})$: $\varepsilon_K=(27+\sqrt{733})/2$, $N(\varepsilon_K)=-1$

737:

20.04194992212249218699136732710848330178755039396965700500970357546492199078823692730371047524302528

comment: $\mathbb{Q}(\sqrt{737})$: $\varepsilon_K=252975383+9318468\sqrt{737}$, $N(\varepsilon_K)=1$

741:

5.501241550403277374364701065603218632194685344763663248238903285521386777276319932587009323942675354

comment: $\mathbb{Q}(\sqrt{741})$: $\varepsilon_K=(245+9\sqrt{741})/2$, $N(\varepsilon_K)=1$

744:

9.615938800086322081079298750513778829396839175129640661437872214042273354274221346425672031562121461

comment: $\mathbb{Q}(\sqrt{186})$: $\varepsilon_K=7501+550\sqrt{186}$, $N(\varepsilon_K)=1$

745:

17.05567817528142723612500334024887927738337920807808519672870584522725947631299652708581022061638622

comment: $\mathbb{Q}(\sqrt{745})$: $\varepsilon_K=12769001+467820\sqrt{745}$, $N(\varepsilon_K)=1$

748:

8.120885932726251299345221131208806917108975543117470019975288521605056282879046731030997791227682097

comment: $\mathbb{Q}(\sqrt{187})$: $\varepsilon_K=1682+123\sqrt{187}$, $N(\varepsilon_K)=1$

749:

9.468464886218090376890446664957240634167615485167768206251518768496136591071467253621620313411088812

comment: $\mathbb{Q}(\sqrt{749})$: $\varepsilon_K=(12945+473\sqrt{749})/2$, $N(\varepsilon_K)=1$

753:

27.14821922637882732723964445963309183815675937062674131030995428804149650105247539091508463240828745

comment: $\mathbb{Q}(\sqrt{753})$: $\varepsilon_K=308526027863+11243313484\sqrt{753}$, $N(\varepsilon_K)=1$

757:

14.82297638650412503966741941071381016009048293571445899089477048639317739380351605978102625157389248

comment: $\mathbb{Q}(\sqrt{757})$: $\varepsilon_K=1369326+49769\sqrt{757}$, $N(\varepsilon_K)=-1$

760:

11.55254994266106489770172372701161615231979873536915799993071185412852200298443155681405168017893300

comment: $\mathbb{Q}(\sqrt{190})$: $\varepsilon_K=52021+3774\sqrt{190}$, $N(\varepsilon_K)=1$

761:

7.377759298852643724067655840726152766092899719915085671171208608417344811501035156772520593865310943

comment: $\mathbb{Q}(\sqrt{761})$: $\varepsilon_K=800+29\sqrt{761}$, $N(\varepsilon_K)=-1$

764:

16.70521542687273197572078046694365428963558170799344812532626926321437629240934915025183475974806531

comment: $\mathbb{Q}(\sqrt{191})$: $\varepsilon_K=8994000+650783\sqrt{191}$, $N(\varepsilon_K)=1$

769:

38.02721354332979150648814540256607642328072454991598811603510443090042289914090245303869517921897385

comment: $\mathbb{Q}(\sqrt{769})$: $\varepsilon_K=16367374077549540+590222604844777\sqrt{769}$, $N(\varepsilon_K)=-1$

773:

4.934525686268375882788071315078557517505642920375536848002106992939954441267576138557058584431285598

comment: $\mathbb{Q}(\sqrt{773})$: $\varepsilon_K=(139+5\sqrt{773})/2$, $N(\varepsilon_K)=-1$

776:

5.966140164436893445938415331494443549135216361555801742751609920347623579492477064951830785750712327

comment: $\mathbb{Q}(\sqrt{194})$: $\varepsilon_K=195+14\sqrt{194}$, $N(\varepsilon_K)=1$

777:

6.100313924734471376822696384459056978746122865114068405866791543825515760779388111732120918492402654

comment: $\mathbb{Q}(\sqrt{777})$: $\varepsilon_K=223+8\sqrt{777}$, $N(\varepsilon_K)=1$

780:

3.330926552641251950919173755890594341105781942856955244599137412719341597074988539162816466107015232

comment: $\mathbb{Q}(\sqrt{195})$: $\varepsilon_K=14+\sqrt{195}$, $N(\varepsilon_K)=1$

781:

18.72235741855227322101577021049642437510736248282492733956119184126182796372216806160004500348771435

comment: $\mathbb{Q}(\sqrt{781})$: $\varepsilon_K=67606199+2419140\sqrt{781}$, $N(\varepsilon_K)=1$

785:

4.025670415869821743372570539268284488531353400438938566473714083679679347563659573385123113504522869

comment: $\mathbb{Q}(\sqrt{785})$: $\varepsilon_K=28+\sqrt{785}$, $N(\varepsilon_K)=-1$

789:

10.36800742243827535296328702571889172479465957648777648667613773404314112708138435623378350513204857

comment: $\mathbb{Q}(\sqrt{789})$: $\varepsilon_K=(31825+1133\sqrt{789})/2$, $N(\varepsilon_K)=1$

793:

9.080914811581315694496760780399783728123238682432734390294889774652880117779606789360671647879049732

comment: $\mathbb{Q}(\sqrt{793})$: $\varepsilon_K=4393+156\sqrt{793}$, $N(\varepsilon_K)=1$

796:

24.20550213882064775704352437071032701315118745246795019103702982407523388835193789089607106953627569

comment: $\mathbb{Q}(\sqrt{199})$: $\varepsilon_K=16266196520+1153080099\sqrt{199}$, $N(\varepsilon_K)=1$

797:

5.905369272483168129044024223291077628711957089648495923310092948156205646600961652647845670442794750

comment: $\mathbb{Q}(\sqrt{797})$: $\varepsilon_K=(367+13\sqrt{797})/2$, $N(\varepsilon_K)=-1$

805:

7.277247249032659709459941687706076870819715303682422954563695528866765442368076649650632222312517104

comment: $\mathbb{Q}(\sqrt{805})$: $\varepsilon_K=(1447+51\sqrt{805})/2$, $N(\varepsilon_K)=1$

808:

8.745443705438446501693195358428476612922371586870433587650598575193490730593276106062000994492136663

comment: $\mathbb{Q}(\sqrt{202})$: $\varepsilon_K=3141+221\sqrt{202}$, $N(\varepsilon_K)=-1$

809:

27.48911653964390589710516399188179168202228375719685283723768985129073607902081324388143069672819876

comment: $\mathbb{Q}(\sqrt{809})$: $\varepsilon_K=433852026040+15253424933\sqrt{809}$, $N(\varepsilon_K)=-1$

812:

4.736121492758925341780541046968346038593599627775760202490818531774812457464831403212217212612549298

comment: $\mathbb{Q}(\sqrt{203})$: $\varepsilon_K=57+4\sqrt{203}$, $N(\varepsilon_K)=1$

813:

8.374246128858244439278464760993957995744697688277393309524540078659853569753983554750608745086406092

comment: $\mathbb{Q}(\sqrt{813})$: $\varepsilon_K=2167+76\sqrt{813}$, $N(\varepsilon_K)=1$

817:

6.530875502754173900245219122551595894084007124882577620381968589717428293305390733457514871650132754

comment: $\mathbb{Q}(\sqrt{817})$: $\varepsilon_K=343+12\sqrt{817}$, $N(\varepsilon_K)=1$

821:

9.692087282057080448114468128309021855198688686596260347742428738232620592258406044705551434893720954

comment: $\mathbb{Q}(\sqrt{821})$: $\varepsilon_K=(16189+565\sqrt{821})/2$, $N(\varepsilon_K)=-1$

824:

11.68746683437468727427771770074011169636401314845919366042983802076832618976404163519265605434947610

comment: $\mathbb{Q}(\sqrt{206})$: $\varepsilon_K=59535+4148\sqrt{206}$, $N(\varepsilon_K)=1$

829:

17.24880603939321356603045941854995756920749254203759209014861012040719174381817048810011563224890386

comment: $\mathbb{Q}(\sqrt{829})$: $\varepsilon_K=15489282+537965\sqrt{829}$, $N(\varepsilon_K)=-1$

840:

4.060145612748403260024260143424263314993586782700685278287879438886600155574458793714960511588252271

comment: $\mathbb{Q}(\sqrt{210})$: $\varepsilon_K=29+2\sqrt{210}$, $N(\varepsilon_K)=1$

844:

27.04530804478411925204155108405191239731476563782504820879389629160125623824790057444733038860168740

comment: $\mathbb{Q}(\sqrt{211})$: $\varepsilon_K=278354373650+19162705353\sqrt{211}$, $N(\varepsilon_K)=1$

849:

35.63849121753060268374868226049588303326174164411662374890804309621002931693344229282747449279146174

comment: $\mathbb{Q}(\sqrt{849})$: $\varepsilon_K=1501654712948695+51536656330476\sqrt{849}$, $N(\varepsilon_K)=1$

853:

10.22132291200726877036249606564692296037938352753206735936046305342621362304157779594813017727650086

comment: $\mathbb{Q}(\sqrt{853})$: $\varepsilon_K=(27483+941\sqrt{853})/2$, $N(\varepsilon_K)=-1$

856:

27.96084154960164191978716800364555524912111786213537108553965230617903096132597452712026730220749559

comment: $\mathbb{Q}(\sqrt{214})$: $\varepsilon_K=695359189925+47533775646\sqrt{214}$, $N(\varepsilon_K)=1$

857:

16.60281152246569178049563680515748991024224969781209728064083706576998301196181805654909065728200874

comment: $\mathbb{Q}(\sqrt{857})$: $\varepsilon_K=8118568+277325\sqrt{857}$, $N(\varepsilon_K)=-1$

860:

4.477207657226921641077831118221195831252415108047527041545103747509502510502892839624766673375167635

comment: $\mathbb{Q}(\sqrt{215})$: $\varepsilon_K=44+3\sqrt{215}$, $N(\varepsilon_K)=1$

861:

6.934396261816368113846773897385611153897396927704806949807210009958606006411207265522395657927072646

comment: $\mathbb{Q}(\sqrt{861})$: $\varepsilon_K=(1027+35\sqrt{861})/2$, $N(\varepsilon_K)=1$

865:

20.36185141655453033682178088050301727501039180283617488509122641794676418595948981085572051699045040

comment: $\mathbb{Q}(\sqrt{865})$: $\varepsilon_K=348345108+11844089\sqrt{865}$, $N(\varepsilon_K)=-1$

869:

10.80724000730202027436303117095130435904031007582069826699362147786902490184922058375875234975608336

comment: $\mathbb{Q}(\sqrt{869})$: $\varepsilon_K=(49377+1675\sqrt{869})/2$, $N(\varepsilon_K)=1$

872:

6.218604087859090689962397471582181929249675846746873397703609276306491092016872325512124671750079319

comment: $\mathbb{Q}(\sqrt{218})$: $\varepsilon_K=251+17\sqrt{218}$, $N(\varepsilon_K)=-1$

876:

4.997166616875528905837596936391388080121864974449470778701212635814690033333192494966776139240619973

comment: $\mathbb{Q}(\sqrt{219})$: $\varepsilon_K=74+5\sqrt{219}$, $N(\varepsilon_K)=1$

877:

13.08705117606196390953372724481251657832225416782318273013375407093623135552758195985832098589937187

comment: $\mathbb{Q}(\sqrt{877})$: $\varepsilon_K=241326+8149\sqrt{877}$, $N(\varepsilon_K)=-1$

881:

26.08283042140594052328779821151831973929085501130422954204527400784453548921130107431756480821589197

comment: $\mathbb{Q}(\sqrt{881})$: $\varepsilon_K=106316171432+3581882825\sqrt{881}$, $N(\varepsilon_K)=-1$

885:

5.472253019083482680631943025385457522468585240135421268553196047887159100855952096559093921202304034

comment: $\mathbb{Q}(\sqrt{885})$: $\varepsilon_K=119+4\sqrt{885}$, $N(\varepsilon_K)=1$

888:

5.697082225561172729715035783666541575614931336676746457112039169164701452200768781379225897884025777

comment: $\mathbb{Q}(\sqrt{222})$: $\varepsilon_K=149+10\sqrt{222}$, $N(\varepsilon_K)=1$

889:

44.72231508204633891504670709927811695892863001690267535694044021912217748068500780394938362398584190

comment: $\mathbb{Q}(\sqrt{889})$: $\varepsilon_K=13231974717803657215+443786188413453504\sqrt{889}$, $N(\varepsilon_K)=1$

892:

6.104788249916012668021806970766429593562006301276650606396381712621275437175691552428742961329526497

comment: $\mathbb{Q}(\sqrt{223})$: $\varepsilon_K=224+15\sqrt{223}$, $N(\varepsilon_K)=1$

893:

7.741098901163667461586134340118292535145967577536268579906902213133236931420661726065936884673046198

comment: $\mathbb{Q}(\sqrt{893})$: $\varepsilon_K=(2301+77\sqrt{893})/2$, $N(\varepsilon_K)=1$

897:

7.088408081909607251554199041857721891886950044581099172602039696752156018725381967449195500639452812

comment: $\mathbb{Q}(\sqrt{897})$: $\varepsilon_K=599+20\sqrt{897}$, $N(\varepsilon_K)=1$

901:

4.094622224330530569959354769455867629636906913996817421789668965194159213714754621371457637976942236

comment: $\mathbb{Q}(\sqrt{901})$: $\varepsilon_K=30+\sqrt{901}$, $N(\varepsilon_K)=-1$

904:

3.402306645480594493342398325781678627007859313271299478025682446773433540956912824819180729277686529

comment: $\mathbb{Q}(\sqrt{226})$: $\varepsilon_K=15+\sqrt{226}$, $N(\varepsilon_K)=-1$

905:

6.582023220547207483387368114279287638763908551521915108677301389133726974655403969940457886176504041

comment: $\mathbb{Q}(\sqrt{905})$: $\varepsilon_K=361+12\sqrt{905}$, $N(\varepsilon_K)=1$

908:

6.113677285129523066102941792095199702255800964820226514119915144268552455963112129619554222707108299

comment: $\mathbb{Q}(\sqrt{227})$: $\varepsilon_K=226+15\sqrt{227}$, $N(\varepsilon_K)=1$

913:

34.56975989650520340919759546981180182002548645040329787669562098460909876537764337500544995646410489

comment: $\mathbb{Q}(\sqrt{913})$: $\varepsilon_K=515734243080407+17068312251564\sqrt{913}$, $N(\varepsilon_K)=1$

917:

7.074116099227877601362741613944134359482347225460702999387859657981281912722036097480311669628678360

comment: $\mathbb{Q}(\sqrt{917})$: $\varepsilon_K=(1181+39\sqrt{917})/2$, $N(\varepsilon_K)=1$

920:

5.203976496118957401583010832180339055199726714183750161931456660680252362743246376908458253866982015

comment: $\mathbb{Q}(\sqrt{230})$: $\varepsilon_K=91+6\sqrt{230}$, $N(\varepsilon_K)=1$

921:

36.15699869650875957986726371294439353460585178825712183459662932622872112597222322907098629242281534

comment: $\mathbb{Q}(\sqrt{921})$: $\varepsilon_K=2522057712835735+83104627139412\sqrt{921}$, $N(\varepsilon_K)=1$

924:

5.023837235487461117271471520519888249612323847163249215563374678156290233122818320059521210120259357

comment: $\mathbb{Q}(\sqrt{231})$: $\varepsilon_K=76+5\sqrt{231}$, $N(\varepsilon_K)=1$

929:

25.81476917763646986406961831280180728765034526977980445243115578875577057149533597105083046456501101

comment: $\mathbb{Q}(\sqrt{929})$: $\varepsilon_K=81317086468+2667927065\sqrt{929}$, $N(\varepsilon_K)=-1$

933:

11.92189110568116199114100168926359831507244539358714741738586705798287749115479127639540296195792306

comment: $\mathbb{Q}(\sqrt{933})$: $\varepsilon_K=75263+2464\sqrt{933}$, $N(\varepsilon_K)=1$

937:

34.51903622348612083302056133328505959851117883679467956505117126080473340612008905908139956119973541

comment: $\mathbb{Q}(\sqrt{937})$: $\varepsilon_K=490226695010796+16015008052621\sqrt{937}$, $N(\varepsilon_K)=-1$

940:

4.521670408657296942872756902834278936974075504846203559256769186562997123363089495819544287943751966

comment: $\mathbb{Q}(\sqrt{235})$: $\varepsilon_K=46+3\sqrt{235}$, $N(\varepsilon_K)=1$

941:

7.034388706176509946035535760150178131938687099012090956594394134599989495227061893944268518492996552

comment: $\mathbb{Q}(\sqrt{941})$: $\varepsilon_K=(1135+37\sqrt{941})/2$, $N(\varepsilon_K)=-1$

949:

10.39467153698288850494664820623634496636608999922920740894934560815005061056120753199275666645146810

comment: $\mathbb{Q}(\sqrt{949})$: $\varepsilon_K=(32685+1061\sqrt{949})/2$, $N(\varepsilon_K)=-1$

952:

10.05732389541310542070281536779589456533265907949026536966444730115129691145728841497666065841300179

comment: $\mathbb{Q}(\sqrt{238})$: $\varepsilon_K=11663+756\sqrt{238}$, $N(\varepsilon_K)=1$

953:

22.42687318569340757620178656356445866136754966424051615463440213745229758751687019337566556790032993

comment: $\mathbb{Q}(\sqrt{953})$: $\varepsilon_K=2746864744+88979677\sqrt{953}$, $N(\varepsilon_K)=-1$

956:

16.33241962387776215686150195155298304955674050891540478006926419693081470262587722599117811423973455

comment: $\mathbb{Q}(\sqrt{239})$: $\varepsilon_K=6195120+400729\sqrt{239}$, $N(\varepsilon_K)=1$

957:

3.432944993774060974261376427383123822591875606794833668041106366200160132486613789345061585729544775

comment: $\mathbb{Q}(\sqrt{957})$: $\varepsilon_K=(31+\sqrt{957})/2$, $N(\varepsilon_K)=1$

965:

3.435026166738481885758424813020385043829826811830143764574290034463055358555823638767888843259401452

comment: $\mathbb{Q}(\sqrt{965})$: $\varepsilon_K=(31+\sqrt{965})/2$, $N(\varepsilon_K)=-1$

969:

17.11791477459839832847420846985873986068149018381181501061328286051140835206036288476103352820239527

comment: $\mathbb{Q}(\sqrt{969})$: $\varepsilon_K=13588951+436540\sqrt{969}$, $N(\varepsilon_K)=1$

973:

14.40687193706628632708133235230008009566953295251861912705945552094994540982879274120085418143545375

comment: $\mathbb{Q}(\sqrt{973})$: $\varepsilon_K=903223+28956\sqrt{973}$, $N(\varepsilon_K)=1$

977:

23.41474602626433220371489990391661753211403139353812562847911672349973764920973838735540276810565106

comment: $\mathbb{Q}(\sqrt{977})$: $\varepsilon_K=7376748868+236003105\sqrt{977}$, $N(\varepsilon_K)=-1$

984:

12.08734541422967205350775246635186853351477379509372736696543978661166766245953269701576898628214330

comment: $\mathbb{Q}(\sqrt{246})$: $\varepsilon_K=88805+5662\sqrt{246}$, $N(\varepsilon_K)=1$

985:

6.704415856786944481961273514495958892163132192962876028548943750172142314291993870515245268553179922

comment: $\mathbb{Q}(\sqrt{985})$: $\varepsilon_K=408+13\sqrt{985}$, $N(\varepsilon_K)=-1$

988:

12.04698312297170429166554752749091682907048780392064820565153858018932760959735004389459829039574447

comment: $\mathbb{Q}(\sqrt{247})$: $\varepsilon_K=85292+5427\sqrt{247}$, $N(\varepsilon_K)=1$

989:

11.54486008340676149342671245792973912543389630770708876076480602672527015915951763218971764583552482

comment: $\mathbb{Q}(\sqrt{989})$: $\varepsilon_K=(103245+3283\sqrt{989})/2$, $N(\varepsilon_K)=1$

993:

8.574329347106446536025671011110736057826861549997995051943172541091453973910555839836832253341512767

comment: $\mathbb{Q}(\sqrt{993})$: $\varepsilon_K=2647+84\sqrt{993}$, $N(\varepsilon_K)=1$

997:

12.04244722177505040900068030510585788899325784721241420466781528393645290893023385544257763419595021

comment: $\mathbb{Q}(\sqrt{997})$: $\varepsilon_K=84906+2689\sqrt{997}$, $N(\varepsilon_K)=-1$

Definition

Let $K=\mathbb{Q}(\sqrt{d})$, with $d>1$ squarefree, be the real quadratic field of fundamental discriminant $D$ and $\varepsilon_K>1$ the fundamental unit of its ring of integers $\mathcal{O}_K$. Listed is the regulator $R_K=\log\varepsilon_K$.

Parameters

$D$

— fundamental discriminant of $K$ ($D$ a fundamental discriminant, $D>1$)

Formulas

(1)

$\kappa_D=\frac{2\,h_K R_K}{\sqrt{D}}$, where $h_K$ is the class number and $\kappa_D=L(1,\chi_D)$ the residue of the Dedekind zeta function of $K$ at $s=1$, listed with $h_K$ in the table of residues.

(2)

$R_K=\operatorname{arcosh}(t/2)$ if $N(\varepsilon_K)=1$ and $R_K=\operatorname{arsinh}(t/2)$ if $N(\varepsilon_K)=-1$, where $t=\varepsilon_K+\bar{\varepsilon}_K=a$ is the trace of $\varepsilon_K=(a+b\sqrt{d})/2$: from $a^2-db^2=\pm 4$, $\varepsilon_K=\bigl(a+\sqrt{a^2\mp 4}\bigr)/2$.

(3)

The fundamental solution of Pell's equation $x^2-dy^2=1$ is $x_1+y_1\sqrt{d}=\varepsilon_K^{k}$ with $k=1$ if $N(\varepsilon_K)=1$ and $\varepsilon_K\in\mathbb{Z}[\sqrt{d}]$, $k=2$ if $N(\varepsilon_K)=-1$ and $\varepsilon_K\in\mathbb{Z}[\sqrt{d}]$, and $k=3$ or $k=6$ respectively when $\varepsilon_K$ has half-integral coordinates; so $\log(x_1+y_1\sqrt{d})=k\,R_K$. For $D=61$, $\varepsilon_K=(39+5\sqrt{61})/2$ has norm $-1$ and half-integral coordinates, and $\varepsilon_K^6=1766319049+226153980\sqrt{61}$ is the famous fundamental solution of $x^2-61y^2=1$.

Comments

(4)

$\varepsilon_K$ is the fundamental unit of the maximal order $\mathcal{O}_K$, not of the order $\mathbb{Z}[\sqrt{d}]$ where $K=\mathbb{Q}(\sqrt{d})$ with $d$ squarefree, and it need not be the smallest solution of a Pell equation. The two differ exactly when $D\equiv 5 \pmod 8$ and $\varepsilon_K$ has half-integral coordinates: $\varepsilon_K=(1+\sqrt{5})/2$ for $D=5$, whereas $2+\sqrt{5}$, the smallest unit of $\mathbb{Z}[\sqrt{5}]$, is $\varepsilon_K^3$. That happens for 77 of the 302 fields here. The comment on each entry gives $\varepsilon_K$ exactly, as $(a+b\sqrt{d})/2$ with $a^2-db^2=\pm 4$, and its norm.

(5)

$\varepsilon_K$ is normalised to exceed $1$; its conjugates $-\varepsilon_K$, $1/\varepsilon_K$ and $-1/\varepsilon_K$ generate the same unit group. $R_K$ is $\log\varepsilon_K$ with no other factor, the convention of the LMFDB and the one under which the class number formula $\kappa_D=2h_KR_K/\sqrt{D}$ holds as written; every real quadratic field has unit rank $1$, so the regulator is a single logarithm rather than a determinant.

(6)

$N(\varepsilon_K)=-1$ for 114 of the 302 fields and $+1$ for the other 188. The norm decides the narrow class number: $h_K^+=h_K$ when $N(\varepsilon_K)=-1$ and $h_K^+=2h_K$ when $N(\varepsilon_K)=1$. Norm $-1$ requires every odd prime dividing $D$ to be $1 \bmod 4$, and is not guaranteed by it: $D=136=8\cdot 17$ has $\varepsilon_K=35+6\sqrt{34}$ of norm $1$.

(7)

$D$ is the discriminant of $K$, not the squarefree integer $d$ with $K=\mathbb{Q}(\sqrt{d})$: $D=d$ if $d\equiv 1 \pmod 4$ and $D=4d$ otherwise, so $\mathbb{Q}(\sqrt{3})$ is $D=12$. The comment on each entry names the field. Every real fundamental discriminant with $D\leq 1000$ is listed, 302 of them, the same enumeration as the residues at $s=1$.

(8)

Only real quadratic fields: an imaginary quadratic field has a finite unit group and its regulator is $1$ by convention.

(9)

$R_K$ is transcendental, being the logarithm of an algebraic number other than $0$ and $1$, but a few entries are constants a reader may hold under another name: $R_K=\log\varphi$ for $D=5$, with $\varphi$ the golden ratio, $\log(1+\sqrt{2})=\operatorname{arsinh}(1)$ for $D=8$, $\log(2+\sqrt{3})=\operatorname{arcosh}(2)$ for $D=12$. In general $R_K=\operatorname{arcosh}(t/2)$ or $\operatorname{arsinh}(t/2)$, where $t$ is the trace of $\varepsilon_K$.

(10)

The regulator of an elliptic curve, held in the tables of regulators of elliptic curves, is the determinant of the canonical height pairing on the Mordell–Weil group, a different object under the same name.

Programs

(P1)

Sage

K.<w> = QuadraticField(61)
eps = K.units(proof=True)[0]                  # a generator: eps, -eps, 1/eps or -1/eps
abs(log(abs(RealField(400)(eps))))            # 3.6642184608864375259...; K.regulator() is the 53-bit value

(P2)

PARI/GP

default(realprecision, 60); bnfinit(x^2 - 61).reg    \\ 3.66421846088643752592584648846429273478387712410404090314775

Links

[1]

Wikipedia: Dirichlet's unit theorem

[2]

Wikipedia: Fundamental unit (number theory)

[3]

Wikipedia: Pell's equation

[4]

LMFDB: real quadratic number fields

[5]

LMFDB: regulator of a number field

[6]

OEIS A014000: first coordinate of the fundamental unit of the real quadratic field of discriminant A003658(n)

[7]

OEIS A014046: its second coordinate

[8]

OEIS A014077: its norm

[9]

OEIS A003658: fundamental discriminants of real quadratic fields

Similar tables

Residues of Dedekind zeta functions of quadratic fields — $\kappa_D=2h_KR_K/\sqrt{D}$ for the same fields; its entry comments carry $h_K$ and $\varepsilon_K$

Golden ratio — $\varphi=\varepsilon_K$ for $D=5$, so $R_K=\log\varphi$ there

Algebraic numbers of degree 2 — holds the units $\varepsilon_K$ themselves for the smallest fields, as roots of $x^2-ax\pm 1$

Regulators of elliptic curves over $\mathbb{Q}$ of rank 1 — a different regulator, of the height pairing on an elliptic curve; likewise the tables for rank 2 and rank 3

$j$-invariants of elliptic curves over quadratic fields with everywhere good reduction — indexed by the same discriminant $D$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (every real fundamental discriminant with $D\leq 1000$ is here)

How they were obtained:

$\varepsilon_K=(a+b\sqrt{d})/2$ is taken exactly, as integers $a,b$, from Sage's unit group with proof=True, and $R_K$ is arb's logarithm of the real ball $(a+b\sqrt{d})/2$ at 397 bits; the digits written are those the ball supports (the worst entry, $D=889$, supports 117). $K.\mathrm{regulator}()$, a 53-bit float, is not used.

Before its logarithm is taken every unit is required to be the smallest unit of $\mathbb{Z}[\sqrt{d}]$ from the continued fraction of $\sqrt{d}$, or its cube root, a computation sharing no code with the unit group. All 302 units were also compared with OEIS A014000/A014046/A014077, eight regulators with their LMFDB pages, and all 302 with the residue table through the class number formula, with the controls that must fail failing.