Class polynomials of $\gamma_2=\sqrt[3]{j}$

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Polynomials

$\Delta$

$P_\Delta^{\gamma_2}(x)$

-4:

x - 12

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

-7:

x + 15

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

-8:

x - 20

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

-11:

x + 32

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

-16:

x - 66

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

-19:

x + 96

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

-20:

x^2 - 100*x - 880

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

-23:

x^3 + 155*x^2 + 650*x + 23375

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

-28:

x - 255

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

-31:

x^3 + 342*x^2 + 837*x + 116127

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

-32:

x^2 - 380*x + 2300

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

-35:

x^2 + 480*x - 5120

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

-40:

x^2 - 780*x + 20880

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

-43:

x + 960

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

-44:

x^3 - 1024*x^2 - 15840*x - 86768

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

-47:

x^5 + 1320*x^4 + 12100*x^3 + 1927375*x^2 + 13571250*x + 252209375

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

-52:

x^2 - 1860*x - 82800

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

-55:

x^4 + 2355*x^3 - 8370*x^2 + 5553900*x - 26484975

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

-56:

x^4 - 2584*x^3 + 133552*x^2 - 722304*x + 21590272

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

-59:

x^3 + 3136*x^2 + 68608*x + 720896

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

-64:

x^2 - 4344*x - 19458

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

-67:

x + 5280

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

-68:

x^4 - 5580*x^3 - 268800*x^2 + 9928000*x - 127840000

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

-71:

x^7 + 6745*x^6 - 327467*x^5 + 51857115*x^4 - 2319299751*x^3 + 41264582513*x^2 - 307873876442*x + 903568991567

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

-76:

x^3 - 9216*x^2 - 47520*x - 938736

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

-79:

x^5 + 11037*x^4 + 170811*x^3 + 123259536*x^2 + 1755868833*x + 17606739231

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

-80:

x^4 - 11660*x^3 - 341220*x^2 + 333200*x + 75024400

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

-83:

x^3 + 13920*x^2 + 128000*x + 8192000

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

-88:

x^2 - 18600*x + 2509200

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

-91:

x^2 + 21792*x - 156672

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

-92:

x^3 - 23035*x^2 + 215050*x - 18437375

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

-95:

x^8 + 27155*x^7 + 1863870*x^6 + 794446725*x^5 + 50622270500*x^4 + 1601654829875*x^3 + 23887845603000*x^2 + 198153584060000*x + 475911004500625

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

-100:

x^2 - 35124*x - 6635376

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

-103:

x^5 + 41250*x^4 - 789525*x^3 + 1709812125*x^2 - 34307398125*x + 306618159375

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

-104:

x^6 - 43712*x^5 + 11405056*x^4 - 541625984*x^3 + 11445137408*x^2 - 17604460544*x + 402971987968

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

-107:

x^3 + 50560*x^2 - 3532800*x + 69632000

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

-112:

x^2 - 65040*x + 1101825

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

-115:

x^2 + 75360*x + 506880

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

-116:

x^6 - 78740*x^5 - 29670224*x^4 - 2334963584*x^3 - 70561570048*x^2 - 778415100928*x - 4652860862464

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

-119:

x^10 + 91451*x^9 - 57318*x^8 + 8544214952*x^7 - 16702428245*x^6 + 16575954139027*x^5 - 279338847074297*x^4 + 3954046021970044*x^3 - 15811869926139793*x^2 + 712791341017769354*x - 2268241533239724383

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

-124:

x^3 - 115974*x^2 + 307989*x - 8432127

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

-127:

x^5 + 133455*x^4 + 3740400*x^3 + 17857891125*x^2 + 481204951875*x + 6837984309375

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

-128:

x^4 - 139760*x^3 - 112700*x^2 - 329092000*x - 7016042500

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

-131:

x^5 + 160512*x^4 + 10305536*x^3 + 238452736*x^2 - 7539261440*x + 52479131648

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

-136:

x^4 - 201684*x^3 + 86773248*x^2 + 1541833920*x + 13430976768

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

-139:

x^3 + 230112*x^2 + 2350080*x + 40697856

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

-140:

x^6 - 240640*x^5 + 25976800*x^4 - 1334584800*x^3 + 24322944000*x^2 - 597047616000*x + 6238797472000

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

-143:

x^10 + 274395*x^9 - 25142150*x^8 + 75587162000*x^7 - 6956035068125*x^6 + 74450728996875*x^5 + 5665559777421875*x^4 + 154537113547265625*x^3 - 897967785317968750*x^2 - 14760295513648437500*x - 52763467060068359375

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

-148:

x^2 - 340440*x - 199148400

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

-151:

x^7 + 387717*x^6 - 21859929*x^5 + 150947885286*x^4 - 8634713921559*x^3 + 233166741715089*x^2 - 2882354303947509*x + 14841592796095983

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

-152:

x^6 - 405120*x^5 + 199795200*x^4 + 34161392000*x^3 + 1524986880000*x^2 + 6331494400000*x + 77881408000000

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

-155:

x^4 + 459360*x^3 + 17966080*x^2 + 2886860800*x + 33449574400

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

-160:

x^4 - 565860*x^3 - 5031180*x^2 + 254653200*x - 664815600

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

-163:

x + 640320

comment: $h(\Delta)=1$; fundamental discriminant of $\mathbb{Q}(\sqrt{-163})$

-164:

x^8 - 666292*x^7 - 528694128*x^6 + 8093909632*x^5 - 4202492559872*x^4 - 23605131515904*x^3 - 884906241454080*x^2 - 6431958836527104*x - 94824650623614976

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

-167:

x^11 + 753825*x^10 + 80729425*x^9 + 571427693250*x^8 + 60142392691250*x^7 + 2458105234231250*x^6 - 102818203422953125*x^5 + 3301150886895390625*x^4 + 12535249981600390625*x^3 + 676382921139277343750*x^2 - 81337944397695312500*x + 31188442383937255859375

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

-172:

x^3 - 921600*x^2 - 475200*x - 884790000

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

-175:

x^6 + 1038375*x^5 + 50719500*x^4 + 1079369148510*x^3 + 51587686215000*x^2 + 1242012635360640*x + 3000640176377505

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

-176:

x^6 - 1080352*x^5 + 177422336*x^4 - 11962091216*x^3 - 5159822304*x^2 + 4846058241024*x + 99474414071824

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

-179:

x^5 + 1215168*x^4 - 229604352*x^3 + 11359879168*x^2 + 89195020288*x + 4108807307264

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

-184:

x^4 - 1477272*x^3 + 1806921648*x^2 - 16946679168*x + 485694206208

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

-187:

x^2 + 1656480*x - 15667200

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

-188:

x^5 - 1720840*x^4 - 374119900*x^3 - 21720325375*x^2 + 185233991250*x - 499123109375

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

-191:

x^13 + 1929156*x^12 + 132432554*x^11 + 3753899991642*x^10 + 252453505687211*x^9 + 62484317128776409*x^8 + 1353022855520958821*x^7 + 11236301719256050633*x^6 - 1334183168583639848979*x^5 + 23935634372069295787010*x^4 - 22786224703538162066266*x^3 + 2707306791349506772875108*x^2 - 44283867777294768088428424*x + 387657998638552482940151327

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

-196:

x^4 - 2327016*x^3 - 3622965840*x^2 - 107185541760*x - 1282205396736

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

-199:

x^9 + 2603907*x^8 - 104566626*x^7 + 6783520510557*x^6 - 279067491990252*x^5 + 8026338875023512*x^4 + 1092866430721131*x^3 + 4583774133474907041*x^2 + 11049573350283305211*x + 182453173698107021391

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

-200:

x^6 - 2704000*x^5 + 4623424000*x^4 - 352921370240*x^3 + 91717084160000*x^2 - 1640953799147520*x + 10444483792506880

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

-203:

x^4 + 3018720*x^3 + 683545600*x^2 + 66412544000*x + 317194240000

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

-208:

x^4 - 3623340*x^3 + 117431100*x^2 + 1540026000*x + 11353770000

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

-211:

x^3 + 4038624*x^2 - 138322944*x + 1744699392

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

-212:

x^6 - 4184940*x^5 - 7423565600*x^4 + 2203394216000*x^3 - 276497126720000*x^2 + 6199864528000000*x - 40706235328000000

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

-215:

x^14 + 4661500*x^13 - 1786135770*x^12 + 21972671494650*x^11 - 8360118862282375*x^10 + 1125083829763489500*x^9 - 54189289821096370375*x^8 + 825370890009985787500*x^7 + 32123749826015269413750*x^6 - 921318927230316747059375*x^5 - 1387616361349727129046875*x^4 + 184862118058544139186109375*x^3 - 653791152747532635789906250*x^2 - 12558458871325161040507812500*x + 76611287379414493935664140625

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

-220:

x^4 - 5567475*x^3 + 74330190*x^2 + 500836500*x + 16120865025

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

-223:

x^7 + 6187290*x^6 + 522296775*x^5 + 38303988817500*x^4 + 3193535171491875*x^3 + 135794984885381250*x^2 + 1619305589395828125*x + 13761937575362109375

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

-224:

x^8 - 6407368*x^7 - 2409043208*x^6 - 277656074368*x^5 - 4870004985296*x^4 + 65996053404160*x^3 - 5230208209338496*x^2 + 56576728224834048*x + 165991267986505984

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

-227:

x^5 + 7114400*x^4 + 479539200*x^3 + 263917568000*x^2 + 186777600000*x + 17196646400000

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

-232:

x^2 - 8459340*x + 24591258000

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

-235:

x^2 + 9372000*x + 228602880

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

-236:

x^9 - 9697280*x^8 + 184045632*x^7 - 444457842896*x^6 - 22885627360256*x^5 - 1865331205175296*x^4 - 78664980351229184*x^3 - 1806502816185008128*x^2 - 17794089344760791040*x - 80827404806057947136

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

-239:

x^15 + 10738244*x^14 + 4722610986*x^13 + 116257782176488*x^12 + 50671255701182371*x^11 + 10232692007537994877*x^10 + 817083452271362573578*x^9 + 40315133572508676197828*x^8 + 1087056980006457751864249*x^7 + 27612005512570024843884125*x^6 + 339788925886840330570075983*x^5 + 6418276739840623903682299500*x^4 + 40849260436933541514555798398*x^3 + 689894909929490622325117277736*x^2 + 586358980345383843347657322144*x + 9449331107481278755944922929311

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

-244:

x^6 - 12704544*x^5 - 44952036480*x^4 + 1236995376768*x^3 - 42250341353472*x^2 + 413858236391424*x - 2141080092831744

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

-247:

x^6 + 14048055*x^5 - 1343241900*x^4 + 197413987600125*x^3 - 19067951072716875*x^2 + 910054436423343750*x - 7412835620267859375

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

-248:

x^8 - 14527720*x^7 + 60248288400*x^6 - 19341012112000*x^5 + 2544071009920000*x^4 + 107512407187200000*x^3 + 1908581616704000000*x^2 + 19222182479360000000*x + 109791608857600000000

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

-251:

x^7 + 16041632*x^6 - 8094431232*x^5 + 1257393586176*x^4 - 33254008684544*x^3 + 257375881658368*x^2 + 2716035312517120*x + 124667644879044608

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

-256:

x^4 - 18909120*x^3 - 2115244152*x^2 - 72777744864*x - 1021025075202

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

-259:

x^4 + 20853888*x^3 + 575106048*x^2 + 21454848000*x + 163669082112

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

-260:

x^8 - 21537680*x^7 - 110728609120*x^6 - 51252692147200*x^5 - 6442753463840000*x^4 + 182651449124864000*x^3 - 467122040265728000*x^2 + 67490598780887040000*x - 148924867991920640000

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

-263:

x^13 + 23740450*x^12 + 309718675*x^11 + 565455557172625*x^10 + 6627349045394375*x^9 + 43853527348980703125*x^8 - 3843213066608663906250*x^7 + 186642345497382912031250*x^6 + 1768168367723447525781250*x^5 + 40409138267859441878906250*x^4 - 270493201895799420732421875*x^3 + 14596473280113361336132812500*x^2 + 160031069424617738283203125000*x + 1088076660345692861859130859375

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

-268:

x^3 - 27878400*x^2 - 2613600*x - 147198006000

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

-271:

x^11 + 30677754*x^10 + 914801751*x^9 + 941124005908965*x^8 + 27116846193913605*x^7 + 9502632858772089*x^6 - 185681845526305239162*x^5 + 9619514682758392508271*x^4 - 170237780630777924156889*x^3 + 1370880271917413462710188*x^2 - 6356983904960317262644455*x + 25341779750946434234523423

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

-272:

x^8 - 31668420*x^7 + 6087487300*x^6 - 3476781866000*x^5 + 263526289640000*x^4 - 2070854383400000*x^3 + 24813652617000000*x^2 - 145350993380000000*x + 2100365289700000000

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

-275:

x^4 + 34824128*x^3 + 17274133504*x^2 + 3639203397632*x - 15853760282624

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

-280:

x^4 - 40756440*x^3 + 262062114480*x^2 - 12346535203200*x + 121082087174400

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

-283:

x^3 + 44749440*x^2 - 117504000*x + 5861376000

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

-284:

x^7 - 46163449*x^6 + 25740303381*x^5 - 6190196251691*x^4 + 72744766970361*x^3 - 5749362548639073*x^2 - 67311863299454554*x - 892037832546317567

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

-287:

x^14 + 50659665*x^13 - 28129021975*x^12 + 2568919855739125*x^11 - 1426937587441134375*x^10 + 126204655474240950000*x^9 + 32999335277874554843750*x^8 + 3027329446774541643593750*x^7 + 33530193906296750589062500*x^6 - 1614538314266736482357421875*x^5 - 35390016191023416075849609375*x^4 - 23323347961283323150097656250*x^3 + 6547351740580562042809326171875*x^2 + 101503932450520991851340332031250*x + 545439788615142196740386962890625

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

-292:

x^4 - 59079180*x^3 - 455450342400*x^2 - 9705848760000*x - 72448045920000

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

-295:

x^8 + 64760565*x^7 - 4922007705*x^6 + 4194083824762050*x^5 - 322924869628729875*x^4 + 9588663838867615875*x^3 + 1391200183347109257375*x^2 + 17701139852322783986250*x + 93682812895759562000625

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

-296:

x^10 - 66770904*x^9 + 490626836656*x^8 + 434315611484160*x^7 + 114794959762763008*x^6 + 6074260303878686720*x^5 + 166936790857906802688*x^4 + 2490653633095245234176*x^3 + 59447014954413364281344*x^2 + 317222930221555706757120*x + 615629953118233965887488

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

-299:

x^8 + 73129344*x^7 + 9881321472*x^6 + 12755115704320*x^5 - 121511241318400*x^4 + 12568076109217792*x^3 - 501350850636021760*x^2 + 6528733126124896256*x - 26339667458536767488

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

Definition

For an admissible $\Delta$, let $H_\Delta$ be the Hilbert class polynomial of $\mathcal{O}_\Delta$ and let $j$ be the modular $j$-invariant. The $\gamma_2$ class polynomial $P_\Delta^{\gamma_2}(x)$ is the unique monic factor of $H_\Delta(x^3)$ in $\mathbb{Z}[x]$ of degree $\deg H_\Delta$, where $\gamma_2^3=j$.

Parameters

$\Delta$

— discriminant ($\Delta<0$, $\Delta\equiv0,1 \pmod 4$, and $3\nmid\Delta$)

Formulas

(1)

$H_\Delta(x^3)=P_\Delta^{\gamma_2}(x)\,Q_\Delta(x)$, where $Q_\Delta(x)=P_\Delta^{\gamma_2}(\zeta_3x)\,P_\Delta^{\gamma_2}(\zeta_3^2x)$ and $\zeta_3$ is a primitive cube root of unity.

Comments

(2)

In the complex multiplication method [3], the condition $3\nmid\Delta$ is the one under which $\gamma_2=\sqrt[3]{j}$ gives a class invariant for $\mathcal{O}_\Delta$, where $j$ is the modular $j$-invariant [2] [1]. The factor $Q_\Delta$ has the roots obtained by multiplying each root of $P_\Delta^{\gamma_2}$ by the two primitive cube roots of unity.

(3)

Non-fundamental discriminants are listed as well as fundamental ones. Each entry's comment gives the class number $h(\Delta)$ and names the order.

(4)

For the discriminants listed here, $P_\Delta^{\gamma_2}$ has the same degree as $H_\Delta$ and defines the same ring class field, while the coefficient height has about one third as many digits, reflecting $\gamma_2^3=j$.

Programs

(P1)

PARI/GP

polclass(-23, 5)

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]

PARI/GP manual: polclass

[3]

Wikipedia: Complex multiplication

Similar tables

Hilbert class polynomials $H_\Delta$ — $P_\Delta^{\gamma_2}$ is the degree $h(\Delta)$ factor of $H_\Delta(x^3)$

Weber class polynomials $W_D$ — store another family of class polynomials with reduced coefficient height, used in the complex multiplication method

$q$-expansion of the $j$-invariant — the modular function whose cube roots give the class invariants here

Rational singular moduli — where $h(\Delta)=1$, $P_\Delta^{\gamma_2}(x)=x-\gamma_2$ and $\gamma_2^3$ is the rational singular modulus for the same $\Delta$

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every discriminant $\Delta<0$, $\Delta\equiv0,1\pmod4$, with $3\nmid\Delta$ and $|\Delta|\leq300$, matching the range of the Hilbert class polynomials)

How they were obtained:

For each $\Delta$, the generator computes the exact Hilbert class polynomial $H_\Delta$ and factors the exact polynomial $H_\Delta(x^3)$ over $\mathbb{Z}$. It selects the unique monic factor of degree $\deg H_\Delta$ and compares the selected polynomial with PARI/GP's polclass(Delta, 5) [2].

This is an exact polynomial computation; no numerical values are stored.