Hilbert class polynomials $H_\Delta$

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Polynomials

$\Delta$

$H_\Delta(x)$

-3:

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-3})$. The value is $x$ because $j(\zeta_3)=0$.

-4:

x - 1728

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(i)$

-7:

x + 3375

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-7})$

-8:

x - 8000

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-2})$

-11:

x + 32768

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-11})$

-12:

x - 54000

comment: $h(\Delta)=1$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-3})$

-15:

x^2 + 191025*x - 121287375

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-15})$

-16:

x - 287496

comment: $h(\Delta)=1$; order of conductor $2$ in $\mathbb{Q}(i)$

-19:

x + 884736

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-19})$

-20:

x^2 - 1264000*x - 681472000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-5})$

-23:

x^3 + 3491750*x^2 - 5151296875*x + 12771880859375

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-23})$

-24:

x^2 - 4834944*x + 14670139392

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-6})$

-27:

x + 12288000

comment: $h(\Delta)=1$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-3})$

-28:

x - 16581375

comment: $h(\Delta)=1$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-7})$

-31:

x^3 + 39491307*x^2 - 58682638134*x + 1566028350940383

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-31})$

-32:

x^2 - 52250000*x + 12167000000

comment: $h(\Delta)=2$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-2})$

-35:

x^2 + 117964800*x - 134217728000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-35})$

-36:

x^2 - 153542016*x - 1790957481984

comment: $h(\Delta)=2$; order of conductor $3$ in $\mathbb{Q}(i)$

-39:

x^4 + 331531596*x^3 - 429878960946*x^2 + 109873509788637459*x + 20919104368024767633

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-39})$

-40:

x^2 - 425692800*x + 9103145472000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-10})$

-43:

x + 884736000

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-43})$

-44:

x^3 - 1122662608*x^2 + 270413882112*x - 653249011576832

comment: $h(\Delta)=3$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-11})$

-47:

x^5 + 2257834125*x^4 - 9987963828125*x^3 + 5115161850595703125*x^2 - 14982472850828613281250*x + 16042929600623870849609375

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-47})$

-48:

x^2 - 2835810000*x + 6549518250000

comment: $h(\Delta)=2$; order of conductor $4$ in $\mathbb{Q}(\sqrt{-3})$

-51:

x^2 + 5541101568*x + 6262062317568

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-51})$

-52:

x^2 - 6896880000*x - 567663552000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-13})$

-55:

x^4 + 13136684625*x^3 - 20948398473375*x^2 + 172576736359017890625*x - 18577989025032784359375

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-55})$

-56:

x^4 - 16220384512*x^3 + 2059647197077504*x^2 + 2257767342088912896*x + 10064086044321563803648

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-14})$

-59:

x^3 + 30197678080*x^2 - 140811576541184*x + 374643194001883136

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-59})$

-60:

x^2 - 37018076625*x + 153173312762625

comment: $h(\Delta)=2$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-15})$

-63:

x^4 + 67515199875*x^3 - 193068841781250*x^2 + 4558451243295023437500*x - 6256903954262253662109375

comment: $h(\Delta)=4$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-7})$

-64:

x^2 - 82226316240*x - 7367066619912

comment: $h(\Delta)=2$; order of conductor $4$ in $\mathbb{Q}(i)$

-67:

x + 147197952000

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-67})$

-68:

x^4 - 178211040000*x^3 - 75843692160000000*x^2 - 318507038720000000000*x - 2089297506304000000000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-17})$

-71:

x^7 + 313645809715*x^6 - 3091990138604570*x^5 + 98394038810047812049302*x^4 - 823534263439730779968091389*x^3 + 5138800366453976780323726329446*x^2 - 425319473946139603274605151187659*x + 737707086760731113357714241006081263

comment: $h(\Delta)=7$; maximal order of $\mathbb{Q}(\sqrt{-71})$

-72:

x^2 - 377674768000*x + 232381513792000000

comment: $h(\Delta)=2$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-2})$

-75:

x^2 + 654403829760*x + 5209253090426880

comment: $h(\Delta)=2$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-3})$

-76:

x^3 - 784074438864*x^2 + 1128678666363648*x - 827237892283232256

comment: $h(\Delta)=3$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-19})$

-79:

x^5 + 1339190283240*x^4 - 6366718450945836*x^3 + 1793441424178093483069839*x^2 - 5859423003994491322155950334*x + 5458041030919737322344464663391

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-79})$

-80:

x^4 - 1597177172000*x^3 - 13028555239824000*x^2 - 171263969177632000000*x + 422286883970526784000000

comment: $h(\Delta)=4$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-5})$

-83:

x^3 + 2691907584000*x^2 - 41490055168000000*x + 549755813888000000000

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-83})$

-84:

x^4 - 3196800946944*x^3 - 5663679223085309952*x^2 + 88821246589810089394176*x - 5133201653210986057826304

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-21})$

-87:

x^6 + 5321761711875*x^5 + 85585228375218750*x^4 + 28321090578679361484375000*x^3 + 497577733884372638735595703125*x^2 + 432181202257616392838287353515625*x + 549806430204864490157810211181640625

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-87})$

-88:

x^2 - 6294842640000*x + 15798135578688000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-22})$

-91:

x^2 + 10359073013760*x - 3845689020776448

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-91})$

-92:

x^3 - 12207823849750*x^2 - 263033266852296875*x - 6267542200571287109375

comment: $h(\Delta)=3$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-23})$

-95:

x^8 + 19874477919500*x^7 - 688170786018119250*x^6 + 395013575867144519258203125*x^5 - 13089776536501963407329479984375*x^4 + 352163322858664726762725228294921875*x^3 - 1437415939871573574572839010971248046875*x^2 + 2110631639116675267953915424764056884765625*x + 107789694576540010002976771996177148681640625

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-95})$

-96:

x^4 - 23340144296736*x^3 + 670421055192156288*x^2 + 447805364111967209472*x - 984163224549635621646336

comment: $h(\Delta)=4$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-6})$

-99:

x^2 + 37616060956672*x - 56171326053810176

comment: $h(\Delta)=2$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-11})$

-100:

x^2 - 44031499226496*x - 292143758886942437376

comment: $h(\Delta)=2$; order of conductor $5$ in $\mathbb{Q}(i)$

-103:

x^5 + 70292286280125*x^4 + 85475283659296875*x^3 + 4941005649165514137656250000*x^2 + 13355527720114165506172119140625*x + 28826612937014029067466156005859375

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-103})$

-104:

x^6 - 82028232174464*x^5 + 739545196164376195072*x^4 + 31013571054009020830449664*x^3 + 1378339984770204584193868955648*x^2 - 25735039642229334200564710375424*x + 65437179730333545242323676123103232

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-26})$

-107:

x^3 + 129783279616000*x^2 - 6764523159552000000*x + 337618789203968000000000

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-107})$

-108:

x^3 - 151013228706000*x^2 + 224179462188000000*x - 1879994705688000000000

comment: $h(\Delta)=3$; order of conductor $6$ in $\mathbb{Q}(\sqrt{-3})$

-111:

x^8 + 236917342626795*x^7 + 12257744369763349962*x^6 + 56129700127461627298044206619*x^5 + 2987537813865962860773420720531252*x^4 - 25675269514993965918445147228203062874*x^3 + 88953282358528708595648019437144660946708*x^2 - 64773995403104720702864091375403035855442761*x + 27524793815819191410861831167197250556510894417

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-111})$

-112:

x^2 - 274917323970000*x + 1337635747140890625

comment: $h(\Delta)=2$; order of conductor $4$ in $\mathbb{Q}(\sqrt{-7})$

-115:

x^2 + 427864611225600*x + 130231327260672000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-115})$

-116:

x^6 - 495202728828032*x^5 - 11056847669496432594944*x^4 - 835102260960042427461140480*x^3 - 66527716583835083670963399688192*x^2 + 143376986667050616958401264069115904*x - 100730316193548175256338136121783353344

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-29})$

-119:

x^10 + 764872171216961*x^9 - 70241355662808988599*x^8 + 585035810262130969538043606647*x^7 - 52855712468679496581065487695942573*x^6 + 4794937071328670764609540039796857947016*x^5 + 12480611255809545689627144542329203076373873*x^4 + 29494022920507896313766601313371285654722780443*x^3 - 292223928830848711011022637790896567674102040378617*x^2 + 346485626218561739292181172729923937711295004460654234*x - 11669920442373800031513478208679663025064587635901689887

comment: $h(\Delta)=10$; maximal order of $\mathbb{Q}(\sqrt{-119})$

-120:

x^4 - 883067971104000*x^3 + 26329406807264910336000*x^2 - 2588458316335175909376000000*x + 4934510722321469030006784000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-30})$

-123:

x^2 + 1354146840576000*x + 148809594175488000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-123})$

-124:

x^3 - 1559739536377947*x^2 - 874125972104525910*x - 599530686551745232383

comment: $h(\Delta)=3$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-31})$

-127:

x^5 + 2375421230598750*x^4 - 30614197896114609375*x^3 + 5642626198092219066070054687500*x^2 - 64331030949386896516600669921875000*x + 319730671478833667491273673675537109375

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-127})$

-128:

x^4 - 2729960418308000*x^3 - 395258439243352250000*x^2 - 55499520947716391500000000*x - 345363656226658026765625000000

comment: $h(\Delta)=4$; order of conductor $4$ in $\mathbb{Q}(\sqrt{-2})$

-131:

x^5 + 4130485792112640*x^4 - 671177121829224448000*x^3 + 107205484283838454093053952*x^2 - 60354680538951673475558801408*x + 144530638394690224075155326369792

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-131})$

-132:

x^4 - 4736863498464000*x^3 - 325211610485778048000000*x^2 + 54984539729717250048000000000*x + 1656636925108948992000000000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-33})$

-135:

x^6 + 7122306993287625*x^5 + 77686119211324699125*x^4 + 50727257383070661992492657625000*x^3 + 628735820731639650833126829398718750*x^2 + 4321223868213674595045534006061857421875*x + 3284527439242119311242957750346113869140625

comment: $h(\Delta)=6$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-15})$

-136:

x^4 - 8151279336430848*x^3 + 735960027609078992953344*x^2 - 1834607111282472051029311488*x + 2422829169428572504087521656832

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-34})$

-139:

x^3 + 12183160834031616*x^2 - 53041786755137667072*x + 67408489017571610198016

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-139})$

-140:

x^6 - 13916138006442400*x^5 - 3270237719203124384000*x^4 - 790870172407252503705600000*x^3 + 2848295663082788926282752000000*x^2 - 96864973869318094511286681600000000*x + 242830180406000275493501698048000000000

comment: $h(\Delta)=6$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-35})$

-143:

x^10 + 20680840776413625*x^9 - 6152626465680933171875*x^8 + 427698864960728554347643841796875*x^7 - 126605537475340807363556327903076171875*x^6 + 34756686762001825564530452864079986572265625*x^5 - 669600645730781825539256319987784782409667968750*x^4 + 6039839065591831041160239782335598758220672607421875*x^3 - 7983490577513055781678247666123524158120155334472656250*x^2 + 172938678455959890794097467818753125064074993133544921875*x - 146892619386926916220452018018596212706528604030609130859375

comment: $h(\Delta)=10$; maximal order of $\mathbb{Q}(\sqrt{-143})$

-144:

x^4 - 23578503968570400*x^3 + 269499185406087942528*x^2 + 490453856866850787293184*x + 571751321233328637579104256

comment: $h(\Delta)=4$; order of conductor $6$ in $\mathbb{Q}(i)$

-147:

x^2 + 34848505552896000*x + 11356800389480448000000

comment: $h(\Delta)=2$; order of conductor $7$ in $\mathbb{Q}(\sqrt{-3})$

-148:

x^2 - 39660183801072000*x - 7898242515936467904000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-37})$

-151:

x^7 + 58309232586862950*x^6 + 1107018219296858557941*x^5 + 3399966616467533664248409155353722*x^4 + 69605133153244389737334180535377491802*x^3 + 779394774943277357155375818745718823538863*x^2 + 271248134304567044479896903675912851345002767*x + 3269200340379000902458720113257045278788199227087

comment: $h(\Delta)=7$; maximal order of $\mathbb{Q}(\sqrt{-151})$

-152:

x^6 - 66246265919280000*x^5 + 17024071380555203520000000*x^4 + 6854544294799483688960000000000*x^3 + 2783058624787093614292992000000000000*x^2 - 1380504171426125758791680000000000000000*x + 472390748138731280269312000000000000000000

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-38})$

-155:

x^4 + 96905542950912000*x^3 - 44477871096357453824000*x^2 + 20396251654725321097216000000*x + 37425860028464856284790784000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-155})$

-156:

x^4 - 109914552886955148*x^3 + 53074935443801207676942*x^2 - 164993592496972989327022035*x + 421266000645144840703921125633

comment: $h(\Delta)=4$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-39})$

-159:

x^10 + 160004137022342145*x^9 + 80544826577049275346921*x^8 + 25601320617862402763638637112400617*x^7 + 12925574698393828230218975528896357700901*x^6 - 500195761724247829095994019904150074287948239*x^5 + 9137627363862789620098285544333489602867649370969*x^4 - 52482545550359592109373005225946991054504656459581968*x^3 + 131220647466890593981458207559990681761779941677658891350*x^2 - 125380849804846892558943693244057944355284981734410556485821*x + 49213884163212475944268367742683491152148907598858879501540753

comment: $h(\Delta)=10$; maximal order of $\mathbb{Q}(\sqrt{-159})$

-160:

x^4 - 181195519824640800*x^3 - 2940735389875294896000*x^2 + 13208221536779701382400000*x - 293835053960432980416000000

comment: $h(\Delta)=4$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-10})$

-163:

x + 262537412640768000

comment: $h(\Delta)=1$; maximal order of $\mathbb{Q}(\sqrt{-163})$; $H_\Delta(0)=640320^3$, the integer that Ramanujan's constant $e^{\pi\sqrt{163}}$ exceeds by $744$ to within $10^{-12}$; the root $j\bigl(\tfrac{1+\sqrt{-163}}{2}\bigr)=-640320^3$

-164:

x^8 - 296853791160440320*x^7 - 161936389233870440957755392*x^6 - 107971538556531472065498397540352*x^5 - 72018009354152588972347870534871023616*x^4 - 82923859178811827895415538602091992842240*x^3 + 58876580988711431943771690012552346623541248*x^2 + 716292304882512928715138362472485709784740265984*x - 852636173252919999445568788749874942641540406706176

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-41})$

-167:

x^11 + 428181809075068500*x^10 - 310443848294435505968750*x^9 + 183339895556073570958521545578125000*x^8 - 132653775309940634844454306979619384765625*x^7 + 99968421621214354876138160879405119659423828125*x^6 + 3228424186003694107655062744056610278450012207031250*x^5 + 54948342744318167377884939629764355959051132202148437500*x^4 - 191958603447999118217843290597001892823611319065093994140625*x^3 + 123751654413478180006143858091723929541723527014255523681640625*x^2 + 41726839319627438364938202440270256635260256938636302947998046875*x + 30337588564062373576333030147629108993519083014689385890960693359375

comment: $h(\Delta)=11$; maximal order of $\mathbb{Q}(\sqrt{-167})$

-168:

x^4 - 483435712076832000*x^3 + 336511679671210230144000000*x^2 - 264691184105480095991808000000000*x + 496644064976895846912000000000000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-42})$

-171:

x^4 + 694282057876537344*x^3 + 472103267541360574464*x^2 + 8391550371275812148084736*x - 1311901521779155773721411584

comment: $h(\Delta)=4$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-19})$

-172:

x^3 - 782759106183330000*x^2 + 1164707517403692000000*x - 692660810326239000000000000

comment: $h(\Delta)=3$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-43})$

-175:

x^6 + 1119444674983992405*x^5 - 54813228576976021387185*x^4 + 1253156381651642217978286627708618800*x^3 - 59496933313401566319649813402788210673425*x^2 + 1368302291061523680379707879639549158890532250*x + 27017288450887144631231387755756779460197062625

comment: $h(\Delta)=6$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-7})$

-176:

x^6 - 1260369120052221040*x^5 - 1311227225704547834164432*x^4 - 1417657940638726253547455241728*x^3 + 56139914410303801525997336800408320*x^2 - 233832181396031563359165936367916838912*x + 984315149136933710414929915123613725364224

comment: $h(\Delta)=6$; order of conductor $4$ in $\mathbb{Q}(\sqrt{-11})$

-179:

x^5 + 1795194552944492544*x^4 - 2200273236852299356176384*x^3 + 2672564790656716736213209317376*x^2 - 23408814596997033103434472837087232*x + 69366107283027836458026686806432415744

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-179})$

-180:

x^4 - 2018504138609120000*x^3 - 2867757758882006477169664000*x^2 + 16660473763558887652278272000000*x - 6776421923145961044033929216000000

comment: $h(\Delta)=4$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-5})$

-183:

x^8 + 2863790268422945625*x^7 + 4218620940804013154578125*x^6 + 8201294924243209292049624110812500000*x^5 + 12093440927683360441327407340610611816406250*x^4 + 655855629401644394905657823337825299835205078125*x^3 + 18238748993199475597203528068101439981700897216796875*x^2 - 26922618461790759850037872887492462842807292938232421875*x + 30451733341148937584624248315225887141980230808258056640625

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-183})$

-184:

x^4 - 3215890895076912384*x^3 + 5767007465145198439020847104*x^2 + 38705419208160503264676104110080*x + 114574710497270997578522590458150912

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-46})$

-187:

x^2 + 4545336381788160000*x - 3845689020776448000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-187})$

-188:

x^5 - 5097838276115828125*x^4 - 8766069746614632866828125*x^3 - 15093437571402131169817626953125*x^2 + 51644103814690479844366699218750*x - 124343484762728525316005706787109375

comment: $h(\Delta)=5$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-47})$

-191:

x^13 + 7178874489555770070*x^12 - 14412129900790076822258611*x^11 + 51536266750679803854633551771840260203*x^10 - 103386239396269087020741974285554789917453508*x^9 + 199518440359885837424153227327826660336955370546108*x^8 - 14768638405830894134972427622171024141266129942650181658*x^7 + 541808230910284083390456000848314355028531599104942728173485*x^6 - 832818220571586800392164744000358270319532131590078379471735694*x^5 + 15253788701960481284921391493158919700255797559466559311296323232498*x^4 - 49375911707911743432917242207615322174325427611237741682026488846612521*x^3 + 14061234326903814621176226216076159779271092623995827091347309743554898127*x^2 + 42312753036411362230230450305478672870803475782951657706005442304434341869586*x + 58256749348304523248144969888837463340160093969294532742146556156000521034802783

comment: $h(\Delta)=13$; maximal order of $\mathbb{Q}(\sqrt{-191})$

-192:

x^4 - 8041801037378436000*x^3 + 15705521635909735050750000*x^2 + 826335556188178615474500000000*x - 1080060886113159937649308593750000

comment: $h(\Delta)=4$; order of conductor $8$ in $\mathbb{Q}(\sqrt{-3})$

-195:

x^4 + 11284411506057216000*x^3 + 25349140792043819237376000*x^2 + 104773100319600336175104000000*x - 233490285492432753672585216000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-195})$

-196:

x^4 - 12626092121367165696*x^3 - 44864481851299856707307347968*x^2 + 250850701957837760512539510177792*x - 2108010653658430719613224868701536256

comment: $h(\Delta)=4$; order of conductor $7$ in $\mathbb{Q}(i)$

-199:

x^9 + 17656190279770938660*x^8 + 1331303100189256816837434*x^7 + 311741055246397228842310784103371345424*x^6 + 23969299805117437326359388515188205981243787*x^5 + 934682848803434155897358662478037099871861466271*x^4 - 15361831050875895680622837467024669907518877308748738*x^3 + 81311504213341585710631261056689664491326495914681965478*x^2 - 26264856563493863087105499097317110823999604480371275106459*x + 6073712999849700354466000422348421795990254023608138785279471

comment: $h(\Delta)=9$; maximal order of $\mathbb{Q}(\sqrt{-199})$

-200:

x^6 - 19733105507276110720*x^5 + 87605036675549431339528253440*x^4 + 236628493411489493484987107953868800*x^3 + 640004261883602853633325553571308188467200*x^2 + 181651879545544879923314552485535205556224000*x + 1139359927820736630329093876556526883461660672000

comment: $h(\Delta)=6$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-2})$

-203:

x^4 + 27502410406723584000*x^3 - 83053272156952592384000000*x^2 + 250634002097696556449792000000000*x + 31913605837856413057024000000000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-203})$

-204:

x^6 - 30703802307926880672*x^5 + 95864841637996112067555072*x^4 + 775121756231241041610849730560*x^3 + 534484930703209896960446929872814080*x^2 + 6020337293681148983229932704488367325184*x + 28508041377034538166862450172153093456658432

comment: $h(\Delta)=6$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-51})$

-207:

x^6 + 42653766018394018375*x^5 - 5002547112103664005187500*x^4 + 1819343755841562591564610147379736328125*x^3 - 210672109851582446065248197114115955810546875*x^2 + 12041028291910181818274355885092809398864746093750*x - 183426864580818496179793649372867188930511474609375

comment: $h(\Delta)=6$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-23})$

-208:

x^4 - 47568078792050004000*x^3 + 4032372412181255526000000*x^2 - 3908668494888708948000000000*x + 1463592841477827633000000000000

comment: $h(\Delta)=4$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-13})$

-211:

x^3 + 65873587288630099968*x^2 + 277390576406111100862464*x + 5310823021408898698117644288

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-211})$

-212:

x^6 - 73387074029381328000*x^5 - 628986407384453487358016000000*x^4 - 2630171369254890916959016960000000000*x^3 - 11008353578715780277672803110912000000000000*x^2 + 39924086528997881772669622484992000000000000000*x - 67450134022842979455115194007552000000000000000000

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-53})$

-215:

x^14 + 101317491792065048950*x^13 - 457954683668826252585187875*x^12 + 10265236278641511113286053799784137196875*x^11 - 46383555189709675449178936346465988262753187500*x^10 + 216305211312669600602461131862011535495470855064453125*x^9 + 31819020878597171136619913529470148872599322438628242187500*x^8 + 2351448778987099675298661052102610490451869537464225291259765625*x^7 + 8679229611593057808179529156250134166357416938123512070321044921875*x^6 - 88614138697852334973112488420763471147133937041181135996684478759765625*x^5 + 110436829236444604799740867885896850392925278958337508378426724884033203125*x^4 + 166187807838211557025418209756031643199878197730031147253756032087707519531250*x^3 + 62715092921872557538045374958431963236662840144482321841785231404293060302734375*x^2 - 612704211737656962652403751307765352670981237392918094651369297974039745330810546875*x + 449653813406963835882488544526476585426974512124916562811345749830665350437164306640625

comment: $h(\Delta)=14$; maximal order of $\mathbb{Q}(\sqrt{-215})$

-216:

x^6 - 112760061542456628096*x^5 + 1197383573848845385478924955648*x^4 - 9056150910670523557375044574248960*x^3 + 129141069874109492050243812631108386816*x^2 + 34441927383131420224661352250608420126720*x + 9316863967448371969962043382716305179148288

comment: $h(\Delta)=6$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-6})$

-219:

x^4 + 155212323706544357376*x^3 + 831039118453558669939310592*x^2 - 15979705448736682450562851012608*x + 110979720274963942538198675506593792

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-219})$

-220:

x^4 - 172572544407169076625*x^3 + 2531540097646020954716625*x^2 - 6015443509589489085440390625*x + 4189527305843979870968496890625

comment: $h(\Delta)=4$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-55})$

-223:

x^7 + 236855705574161972250*x^6 - 43072684586065004589140625*x^5 + 56100625266918564788759127557586158203125*x^4 - 10118589468858067354789356763548186038818359375*x^3 + 904981240117595334764254951261845701135925292968750*x^2 - 3017942224498278012503966427816688964673110961914062500*x + 2606386098587221959562486420442180281995713710784912109375

comment: $h(\Delta)=7$; maximal order of $\mathbb{Q}(\sqrt{-223})$

-224:

x^8 - 263096730270583432768*x^7 - 1723077455096553031935888128*x^6 - 11042808304392149169199170364149760*x^5 - 1547817012108377539813804203697900199936*x^4 - 23454276662221670119469240221271015345094656*x^3 - 134143607306227801938718107847574206949591351296*x^2 + 333907584600306671024017785849535798840531183730688*x + 4573574179879344596745560367912999678227803908603904

comment: $h(\Delta)=8$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-14})$

-227:

x^5 + 360082897644683264000*x^4 - 2562327002832961536000000000*x^3 + 18227340807938993794580480000000000*x^2 - 2111118203460821622718464000000000000*x + 5085472193216544027705344000000000000000

comment: $h(\Delta)=5$; maximal order of $\mathbb{Q}(\sqrt{-227})$

-228:

x^4 - 399605224650084576000*x^3 - 7985216535621460489954944000000*x^2 + 58827548670433207062445836288000000000*x + 120020259495560805847424176128000000000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-57})$

-231:

x^12 + 545412728031003167382*x^11 + 4575214689490213216643494701*x^10 + 297475044886884495086385167578291254071163*x^9 + 2495822670620928334448459105500627052803887158892*x^8 + 542945869903650549618648833813237732507684112466148436*x^7 + 59883227623310630463788192163427650885804413659842024636798*x^6 + 424012974154213345546525745525460576866398466044926984841121493*x^5 + 10526543646171305092444013519332915620006154949645088229399234651338*x^4 - 11279114369323207645565898199805585890834325844520121148272433760751685*x^3 + 87871631779286293777040608764652776494101765559955668704283513742443115297*x^2 + 5622739472203769328992152587174484508607105253856090982259484423919689974446*x - 10274960660991508138072982056831840368163135985652317753453065819164012139188831

comment: $h(\Delta)=12$; maximal order of $\mathbb{Q}(\sqrt{-231})$

-232:

x^2 - 604729957849891344000*x + 14871070713157137145512000000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-58})$

-235:

x^2 + 823177419449425920000*x + 11946621170462723407872000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-235})$

-236:

x^9 - 911900087980611037808*x^8 - 8829246291887799055677742080*x^7 - 85653894916610237425482419166691328*x^6 + 890259336020656022114227027503791079424*x^5 - 2609998050684441805972918666809036652763152384*x^4 + 31142196350994346572482398388463886712080993091584*x^3 - 674608806297136836505485361662125850918390913495465984*x^2 + 618701375850809570249968077855481255020992055993767559168*x - 528051042407719084320821843257565323032509500612428437651456

comment: $h(\Delta)=9$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-59})$

-239:

x^15 + 1238073883315407514496*x^14 - 13666850068795802991492370368*x^13 + 1532827087345492867204119093852378082438678*x^12 - 16918290867054590371279243347863233076912830800656*x^11 + 181721330191211254702985841157524557972842205740584445069*x^10 - 53402042304078887007585772925287830545907353075301922255822235*x^9 + 7812948654819296135589446079820190138351182297783287462126807084194*x^8 - 213242735312342090716157115838233952503532476492926509784331970735677434*x^7 + 2345429910309004121222696836646913020268276389308953560399316037687685228093*x^6 - 3810541708818932268937818042616997315880367489562442234228253037628692447242929*x^5 - 17149947441960712280092489124465992101403607262591607787910136971714285286547327394*x^4 + 6777279017108616482947859571620293678609069493537534791893310613536520130766425993446*x^3 + 208933803735777172616099243225384508569454797521044528682952239050793396911848873404608651*x^2 - 323628361549512079174002784188728374161166898133764200695741531312961505222798905042238214*x + 843729436361519079684848812880135988137701277041482416947907638671428789917777110886130507231

comment: $h(\Delta)=15$; maximal order of $\mathbb{Q}(\sqrt{-239})$

-240:

x^4 - 1370337635584848362400*x^3 + 15510636623637225985530243375*x^2 - 3213137488352330508627918491550000*x + 51848746810441819437662737568196890625

comment: $h(\Delta)=4$; order of conductor $4$ in $\mathbb{Q}(\sqrt{-15})$

-243:

x^3 + 1855762905734664192000*x^2 - 3750657365033091072000000*x + 3338586724673519616000000000

comment: $h(\Delta)=3$; order of conductor $9$ in $\mathbb{Q}(\sqrt{-3})$

-244:

x^6 - 2052295773725248986240*x^5 - 92973717558373200586964869091328*x^4 - 2691275293785918359227938328726732800*x^3 - 31292753080096691789898512325924416913408*x^2 - 24417475317780070950649666808040757791817728*x - 9815190670232173018201554731440614047465078784

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-61})$

-247:

x^6 + 2772410642909877080250*x^5 + 888629892547516768433109375*x^4 + 7686260773063033411724550958439950781250000*x^3 + 2475076441475565987510057965501956793452880859375*x^2 + 400348022833121004028281794619328068026346954345703125*x - 407336295332190846580777495118233696120388820648193359375

comment: $h(\Delta)=6$; maximal order of $\mathbb{Q}(\sqrt{-247})$

-248:

x^8 - 3063517083860376640000*x^7 + 169518269276842782112073472000000*x^6 + 2461626754066908714341150658560000000000*x^5 + 35762831246449484056560587036188672000000000000*x^4 - 27750969592459084458872706174812160000000000000000*x^3 + 271994089256402280576009987295281152000000000000000000*x^2 - 1093432823745012115729536788054671360000000000000000000000*x + 1323449723347621474969758725859966976000000000000000000000000

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-62})$

-251:

x^7 + 4128446190315309498368*x^6 - 66204185373144403998280777728*x^5 + 1062008880270126105976008028408774656*x^4 + 7966552994949346594041401247164174172160*x^3 + 416131608793437401577832999781610387970981888*x^2 - 1791911545705841840084320427251134859220759871488*x + 1937587239465703269672056660685864050152464252403712

comment: $h(\Delta)=7$; maximal order of $\mathbb{Q}(\sqrt{-251})$

-252:

x^4 - 4558302700896532039875*x^3 + 10887176246183122533468750*x^2 - 2443574658947106282398437500*x + 398963060554172791168212890625

comment: $h(\Delta)=4$; order of conductor $6$ in $\mathbb{Q}(\sqrt{-7})$

-255:

x^12 + 6128344649561048076375*x^11 + 109818071224486731519520855500*x^10 + 37556608101691968890237510940495083062359375*x^9 + 673058835898027399768590893111635020590531709609375*x^8 - 257071952051162192994763669120415771028223618153289062500*x^7 + 48780025238715852051357522255186510362783551525181839218750000*x^6 + 1050548488006167152734252478408891496811178954807077033503662109375*x^5 - 2325968187865245555746840267996850910210272748417670038737172851562500*x^4 + 23418529525567894791342125100999050122359225461527148497750530120849609375*x^3 - 30631401929712387173525528417517956965633001886992450471268726906005859375000*x^2 + 8285413263557506490462798160453959059451866515780958833500086781616210937500000*x - 30414382619613446559856313259492414970013542565259802806559309274202724456787109375

comment: $h(\Delta)=12$; maximal order of $\mathbb{Q}(\sqrt{-255})$

-256:

x^4 - 6761166974781862161312*x^3 - 1826592673506207200904172752*x^2 + 26925623396663008311375890966784*x - 1064410681181869521037208505239142408

comment: $h(\Delta)=4$; order of conductor $8$ in $\mathbb{Q}(i)$

-259:

x^4 + 9068999694311625523200*x^3 - 368189472100537894019530752*x^2 + 5493320206929896679139197321216*x + 4384296738486457527093398159228928

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-259})$

-260:

x^8 - 9997874035270492198400*x^7 - 999896161895842101863690217472000*x^6 - 21507054600723946274941348498171494400000*x^5 - 463238908732347767153420578775505775886336000000*x^4 + 14865557804649865113150034077076664167379763200000000*x^3 - 85980083235988029405783249092189509918128078848000000000*x^2 + 305486088367929951707960768526477860306636557516800000000000*x - 3302947505675715028946774256661472679426359558144000000000000

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-65})$

-263:

x^13 + 13380310119466658931625*x^12 - 311327635162565982167010546875*x^11 + 179032706285420766577202734457392432796875000*x^10 - 4165394082070752123229551692087292204450178222656250*x^9 + 98905316949525970240143951409615108996096316126037597656250*x^8 + 47530953928226416306068505409986490840011703185605743408203125000*x^7 + 11458136561160708300394038317028343694717312841071722625732421875000000*x^6 - 43719503450979282495514171655395663354927991909842598140180110931396484375*x^5 - 217712350218237684687170899414325350050388748483588898629844188690185546875000*x^4 + 596774975718198603949702355142508889167524972921148663966583088040351867675781250*x^3 + 6162774782812542130030147316585405729653544328652457605331892962567508220672607421875*x^2 - 4487212542386417035130564561724254020346483123482676773093024865374900400638580322265625*x + 1288185729855109347493715381276499192204085431957219681118500147931626997888088226318359375

comment: $h(\Delta)=13$; maximal order of $\mathbb{Q}(\sqrt{-263})$

-264:

x^8 - 14739806897587232709120*x^7 + 1789885567319176457551625511223296*x^6 - 43813781353344480858785503406172212822016*x^5 - 1576759051947634872250887243973927048713338880*x^4 - 19114071480061200751208790258848908645703796916224*x^3 + 189771022593359719599623857042603680684355481140985856*x^2 - 171749422417263603359883069647289043394815149890303164416*x + 327886345447155202813840576100201205111813144244435638288384

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-66})$

-267:

x^2 + 19683091854079488000000*x + 531429662672621376897024000000

comment: $h(\Delta)=2$; maximal order of $\mathbb{Q}(\sqrt{-267})$

-268:

x^3 - 21667237292024856738000*x^2 + 32240842762858236972000000*x - 3189376432736929569384216000000000

comment: $h(\Delta)=3$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-67})$

-271:

x^11 + 28871507301417512542197*x^10 - 16464632106657057718070503233*x^9 + 833563933860509791181111419288850809747772703*x^8 - 474119236512804370442479006085162435392319850885990*x^7 + 135135894145413545460315236328333854986568618194277465631*x^6 + 1774664025734848444303272990784027252090100477956938931603749*x^5 + 131114086157450672131023977149470819445696255045995630161701416198*x^4 - 270811671641227589728387064696795035502951934500561387965958352794385*x^3 + 402828435183506117139311560601486140106573513556923825752918282230153207*x^2 + 77658572899931472455557097251364224126893796442704184730369602375352670242*x + 16274637962340468994088643348402984244223247201766753166531174639738274287967

comment: $h(\Delta)=11$; maximal order of $\mathbb{Q}(\sqrt{-271})$

-272:

x^8 - 31759326199909682088000*x^7 - 992292506997017283600644000000*x^6 - 31424909544599612739578321240000000000*x^5 + 13460227027917301519133262366182000000000000*x^4 + 3861901192470234862993839021544000000000000000*x^3 - 7711135475352672738710945448900000000000000000000*x^2 - 7751659760972060765625755308904000000000000000000000*x + 9265833623430102037137881938273000000000000000000000000

comment: $h(\Delta)=8$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-17})$

-275:

x^4 + 42230108051959368384512*x^3 - 1470671864720383632491493195776*x^2 + 51213041627075282291106746090041376768*x - 3984711300201636241319486354007863066624

comment: $h(\Delta)=4$; order of conductor $5$ in $\mathbb{Q}(\sqrt{-11})$

-276:

x^8 - 46421840776490384352768*x^7 - 10000254249186327541943559851360256*x^6 + 359684269203806519420937768802377441214464*x^5 - 7594738933192260668250057199680097904322674688*x^4 + 407334970460053160180107543216439671099066662518784*x^3 + 767440733750724125562378402812781678248671343699558400*x^2 - 6523546308273811582074020329970859860219102556953002377216*x + 11579958468886822266431515535986431304105856851373387044356096

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-69})$

-279:

x^12 + 61599632504312736062442*x^11 + 39085157358431990256370399479*x^10 + 3794514724678786084595916293218126177015471815*x^9 + 2413273773047259527195253084202133346123169130971473*x^8 + 767669753103635395090961318968027363505110782876830273133*x^7 - 58283377462504689708116605269364153282329645388892963462482473*x^6 + 1657978149641301982239247099561052459846466574320400009376232822498*x^5 - 10013227239675271500918792405158329997586590051360922807501315554033466*x^4 + 24732016651714724865787524804888521394564660407243437120880676248642113002*x^3 - 26776389980763040000394320636711170871388575884924609791873059552120526971044*x^2 + 10814510784493696645623229836489660032236288090281566851237671363163259870593752*x + 108587827741394171370512478914675458441925050289489507636190825980073172090094721

comment: $h(\Delta)=12$; order of conductor $3$ in $\mathbb{Q}(\sqrt{-31})$

-280:

x^4 - 67667966893419063840000*x^3 + 17602516524144666384420962098176000*x^2 - 708555761206745670461365038563328000000*x + 1775168961518724506399346503073398784000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-70})$

-283:

x^3 + 89611323386832801792000*x^2 + 90839236535446929408000000*x + 201371843156955365376000000000

comment: $h(\Delta)=3$; maximal order of $\mathbb{Q}(\sqrt{-283})$

-284:

x^7 - 98373700603090523199715*x^6 - 4546951697981923176956448676986*x^5 - 209903939873446020495519316596118272758*x^4 - 12543980353722772760179537174109237891242365*x^3 - 282310474518147042483634599702449209357160103606*x^2 + 904330206367752622526324160408849797910089300094965*x - 709822597815616788561460999971455889888479357689323263

comment: $h(\Delta)=7$; order of conductor $2$ in $\mathbb{Q}(\sqrt{-71})$

-287:

x^14 + 130017330972527578537125*x^13 - 6698357657618306959300749765625*x^12 + 16904506692603022625452451650285182736386718750*x^11 - 870877447830448509906310286855016521261913044677734375*x^10 + 44124406751543298486951757064546752620852454933728424072265625*x^9 - 37440626797808974887161698082710365926423579373005642490386962890625*x^8 + 15857859011592456271411600608807407083008200255495359166626930236816406250*x^7 - 77741168106987348796498736408928032449881769854967153651849091053009033203125*x^6 - 689666073544550013966593282843777525224970203263486012610490672290325164794921875*x^5 + 2796907818941754701021666870337999145633786876397964262080491920933127403259277343750*x^4 + 38922739785997546759451891252935269190758546113162117006238185894908383488655090332031250*x^3 - 88571091393369945585027038239709934745086635483510805720054231573652941733598709106445312500*x^2 - 62484801377078374907978416472072998345681955148585098464247473540905048139393329620361328125000*x + 162270825957231643818056928693921849618705394682014151595002795007445683950209058821201324462890625

comment: $h(\Delta)=14$; maximal order of $\mathbb{Q}(\sqrt{-287})$

-288:

x^4 - 142637765058468510772000*x^3 - 87330008255955399131086000000*x^2 + 136478143044657426076564000000000*x + 40994594700208456153393000000000000

comment: $h(\Delta)=4$; order of conductor $6$ in $\mathbb{Q}(\sqrt{-2})$

-291:

x^4 + 188155567079341753466880*x^3 + 10786588141336392324590050738176*x^2 + 285389231946718842181542553187254272*x + 21782000952710117887925312635418808680448

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-291})$

-292:

x^4 - 206287709860428304608000*x^3 - 93693622511929038759497066112000000*x^2 + 45521551386379385369629968384000000000*x - 380259461042512404779990642688000000000000

comment: $h(\Delta)=4$; maximal order of $\mathbb{Q}(\sqrt{-73})$

-295:

x^8 + 271602295664902418108250*x^7 + 289315392383740839332561391375*x^6 + 73767807010599056699488845989197941140892890625*x^5 + 78688417471647009524122019473588050719938852358765625*x^4 + 41940062336716757201181279559382045880935056981401849609375*x^3 + 1811885751513084753220927364888013159601085995323528927843750000*x^2 - 1122282566856104887683461567415963897525574528336611314070556640625*x + 822204343689207610829131926678660596532719371553249925736718994140625

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-295})$

-296:

x^10 - 297590021529144696899712*x^9 + 162320889747737857249685120747130880*x^8 + 10832757073596276258900582218946579850592256*x^7 + 723330380126653611855128035587837975712897397948416*x^6 - 18343602778583830315600785964203789047910620519106347008*x^5 + 1679209288192190324963522812029010220997720672517680940449792*x^4 + 10593212648456356679691765850382563233587415411210119731327533056*x^3 + 109273800094565901214220123670634449029720795923690895083186480480256*x^2 - 74426868895161208470636584499354600919093741197724233674495663013888*x + 233323899475941320904470255549714742154931061785021169977723889513398272

comment: $h(\Delta)=10$; maximal order of $\mathbb{Q}(\sqrt{-74})$

-299:

x^8 + 391086320728105978429440*x^7 - 28635280874816126174326167699456*x^6 + 2094055410006322146651491130721133658112*x^5 - 186547260770756829961971675685151791296544768*x^4 + 6417141278133218665289808655954275181523718111232*x^3 - 19207839443594488822936988943836177115227877227364352*x^2 + 45797528808215150136248975363201860724351225694802411520*x - 18273883965326272223717626628647422907813731016193733558272

comment: $h(\Delta)=8$; maximal order of $\mathbb{Q}(\sqrt{-299})$

-300:

x^6 - 428244362959801779810720*x^5 + 32278855882815402576742692253440*x^4 - 233405320133674124312518469774131200*x^3 + 21122955530832902270001123584504233628467200*x^2 + 62082816308629282586712746552975312469884928000*x + 66661978554978958501295319312489107870472732672000

comment: $h(\Delta)=6$; order of conductor $10$ in $\mathbb{Q}(\sqrt{-3})$

Definition

Let $\Delta<0$ be a discriminant, $\Delta\equiv 0$ or $1 \pmod 4$, and $\mathcal{O}_\Delta$ the imaginary quadratic order of discriminant $\Delta$. The Hilbert class polynomial is $H_\Delta(x)=\prod_{[\mathfrak{a}]}\bigl(x-j(\mathfrak{a})\bigr)$, the product over the $h(\Delta)$ classes of proper $\mathcal{O}_\Delta$-ideals $\mathfrak{a}$ of the $j$-invariants of the lattices $\mathfrak{a}\subset\mathbb{C}$.

Parameters

$\Delta$

— discriminant ($\Delta<0$, $\Delta\equiv 0,1 \pmod 4$)

Formulas

(1)

$H_\Delta(0)$ is a perfect cube whenever $3\nmid\Delta$: then Weber's $\gamma_2=\sqrt[3]{j}$ takes values in the ring class field, so the norm of $j$ is a cube ([1], §12).

(2)

$H_\Delta(x)=\prod_{(a,b,c)}\Bigl(x-j\bigl(\tfrac{-b+\sqrt{\Delta}}{2a}\bigr)\Bigr)$, the product over the reduced primitive positive definite binary quadratic forms $ax^2+bxy+cy^2$ of discriminant $b^2-4ac=\Delta$: $|b|\leq a\leq c$, $\gcd(a,b,c)=1$, and $b\geq 0$ when $|b|=a$ or $a=c$. There are $h(\Delta)$ of them.

(3)

For a prime $\ell$, the diagonal of the modular polynomial factors as $\Phi_\ell(x,x)=-\prod_{\Delta}H_\Delta(x)^{r_\ell(\Delta)}$, with $r_\ell(\Delta)=\#\{\alpha\in\mathcal{O}_\Delta : N(\alpha)=\ell\}/w_\Delta$ and $w_\Delta$ the number of units of $\mathcal{O}_\Delta$; only $\Delta=(t^2-4\ell)/f^2$ with $t^2<4\ell$ occur, and $\sum_\Delta r_\ell(\Delta)\,h(\Delta)=2\ell$ ([1], §13). For $\ell=2$: $\Phi_2(x,x)=-(x-1728)(x+3375)^2(x-8000)$.

(4)

For a prime $p\nmid\Delta$: $4p=X^2-\Delta Y^2$ has a solution in integers if and only if $\left(\frac{\Delta}{p}\right)=1$ and $H_\Delta$ has a root modulo $p$ ([1], Theorem 9.2, stated there for $\Delta=-4n$ and for $p\nmid\operatorname{disc}H_\Delta$, a hypothesis that is automatic once $p$ splits in $K$: two singular moduli of discriminant $\Delta$ coincide modulo $p$ only at supersingular reduction).

Comments

(5)

The coefficients grow like $\exp\bigl(\pi\sqrt{|\Delta|}\sum 1/a\bigr)$, the sum over the reduced forms, and the degree like $\sqrt{|\Delta|}$. The OEIS entry [2] carries the coefficients to $\Delta=-500$, and the Sage and PARI programs here give any further one.

(6)

The roots of $H_\Delta$ are the singular moduli of discriminant $\Delta$, the values $j(\tau)$ at the imaginary quadratic $\tau$ with $\mathrm{End}\langle\tau,1\rangle=\mathcal{O}_\Delta$; they are algebraic integers. The thirteen rows of degree one, where $H_\Delta(x)=x-j$, are exactly the rational singular moduli.

(7)

Every discriminant is listed, not only the fundamental ones. Writing $\Delta=f^2\Delta_0$ with $\Delta_0$ fundamental, $f$ is the conductor of $\mathcal{O}_\Delta$ in $K=\mathbb{Q}(\sqrt{\Delta})$, and the roots of $H_\Delta$ generate the ring class field of $\mathcal{O}_\Delta$ over $K$, which is the Hilbert class field of $K$ exactly when $f=1$. Each entry's comment names the order and its class number $h(\Delta)$, which is the degree.

(8)

This is the polynomial the CM method reduces modulo $p$: for a prime $p$ with $4p=X^2-\Delta Y^2$, the roots of $H_\Delta$ in $\mathbb{F}_p$ are the $j$-invariants of the elliptic curves over $\mathbb{F}_p$ with complex multiplication by $\mathcal{O}_\Delta$, and those curves have $p+1\pm X$ points. That is how curves of prescribed order are built and how the ECPP primality test finds its curves.

(9)

These are the class polynomials of $j$ itself. The Weber class polynomials and the other class polynomials of smaller height, built from Weber's $\mathfrak{f}$ or from $\gamma_2$, are different polynomials with the same splitting fields and are not listed here.

(10)

$H_\Delta$ is monic with integer coefficients, irreducible over $\mathbb{Q}$, and of degree $h(\Delta)$, the class number of $\mathcal{O}_\Delta$.

Programs

(P1)

PARI/GP

polclass(-303)

(P2)

Sage

polynomials = {D: hilbert_class_polynomial(D) for D in [-300..-3] if D % 4 in [0, 1]}
hilbert_class_polynomial(-303)      # the next one after this table

References

[1]

David A. Cox, "Primes of the form $x^2+ny^2$: Fermat, class field theory, and complex multiplication", 2nd edition, Wiley, 2013.

Links

[2]

OEIS A305474: the coefficients of $H_\Delta$, constant term first, $\Delta=-3,-4,-7,\dots$

[3]

SageMath reference: complex multiplication, hilbert_class_polynomial

[4]

Wikipedia: Complex multiplication

Similar tables

Rational singular moduli — the thirteen entries of degree one, $H_\Delta(x)=x-j$

Modular polynomials for the $j$-invariant — $\Phi_\ell(x,x)$ is a product of powers of $H_\Delta$

Ramanujan's constant — $e^{\pi\sqrt{163}}$ is within $10^{-12}$ of $H_{-163}(0)+744=640320^3+744$

$q$-expansion of the $j$-invariant — the function whose values at imaginary quadratic points are the roots

Residues of Dedekind zeta functions of quadratic fields — the same fields, indexed by fundamental discriminant; its comments carry $h(\Delta)$

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

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (every such discriminant with $|\Delta|\leq 300$ is here, 94 fundamental and 56 not; the range is set by how long an entry becomes)

How they were obtained:

Each polynomial is Sage's hilbert_class_polynomial (FLINT, complex interval arithmetic), required by the generator to agree with PARI's polclass, a different algorithm; to have degree equal to the number of reduced primitive forms counted by brute force; and to be enclosed, coefficient by coefficient, in balls of radius below $1/2$ by the product $\prod(x-j(\tau))$ over those forms computed in ComplexBallField at 1500 bits, which determines an integer coefficient.

All 150 were also compared with the 250 rows of the OEIS b-file of A305474. The stored polynomials were also checked against the cube formula on every entry, against the diagonal of the modular polynomials for $j$ for $\ell=2,3,5,7,11$, and against the splitting criterion for every $\Delta$ here and every prime $p<400$, in both directions.