LMFDB - Number field 36.36.42867551195672722495351149174628212645190717310766431451792643783547492533.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279)

gp: K = bnfinit(y^36 - y^35 - 110*y^34 + 110*y^33 + 5551*y^32 - 5551*y^31 - 170273*y^30 + 170273*y^29 + 3546007*y^28 - 3546007*y^27 - 53034356*y^26 + 53034356*y^25 + 587601367*y^24 - 587601367*y^23 - 4903561973*y^22 + 4903561973*y^21 + 31025687812*y^20 - 31025687812*y^19 - 148620561113*y^18 + 148620561113*y^17 + 534035184802*y^16 - 534035184802*y^15 - 1413681908438*y^14 + 1413681908438*y^13 + 2676523987366*y^12 - 2676523987366*y^11 - 3458784856340*y^10 + 3458784856340*y^9 + 2828954020750*y^8 - 2828954020750*y^7 - 1286656880618*y^6 + 1286656880618*y^5 + 256697207395*y^4 - 256697207395*y^3 - 15659396372*y^2 + 15659396372*y - 1324838279, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279)

$ x^{36} - x^{35} - 110 x^{34} + 110 x^{33} + 5551 x^{32} - 5551 x^{31} - 170273 x^{30} + \cdots - 1324838279 $ x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$36$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[36, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$42867551195672722495351149174628212645190717310766431451792643783547492533$ 42867551195672722495351149174628212645190717310766431451792643783547492533 $\medspace = 11^{18}\cdot 37^{35}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$111.00$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$11^{1/2}37^{35/36}\approx 111.00356009125302$
Ramified primes:	$11$, $37$ 11, 37	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{37}) $
$\card{ \Gal(K/\Q) }$:	$36$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$407=11\cdot 37$
Dirichlet character group:	$\lbrace$$\chi_{407}(384,·)$, $\chi_{407}(1,·)$, $\chi_{407}(386,·)$, $\chi_{407}(131,·)$, $\chi_{407}(12,·)$, $\chi_{407}(397,·)$, $\chi_{407}(142,·)$, $\chi_{407}(144,·)$, $\chi_{407}(274,·)$, $\chi_{407}(153,·)$, $\chi_{407}(155,·)$, $\chi_{407}(287,·)$, $\chi_{407}(32,·)$, $\chi_{407}(34,·)$, $\chi_{407}(43,·)$, $\chi_{407}(54,·)$, $\chi_{407}(188,·)$, $\chi_{407}(318,·)$, $\chi_{407}(67,·)$, $\chi_{407}(76,·)$, $\chi_{407}(78,·)$, $\chi_{407}(208,·)$, $\chi_{407}(210,·)$, $\chi_{407}(342,·)$, $\chi_{407}(87,·)$, $\chi_{407}(221,·)$, $\chi_{407}(351,·)$, $\chi_{407}(98,·)$, $\chi_{407}(100,·)$, $\chi_{407}(230,·)$, $\chi_{407}(232,·)$, $\chi_{407}(362,·)$, $\chi_{407}(109,·)$, $\chi_{407}(241,·)$, $\chi_{407}(243,·)$, $\chi_{407}(122,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{64624199}a^{19}-\frac{6183736}{64624199}a^{18}-\frac{57}{64624199}a^{17}+\frac{10800749}{64624199}a^{16}+\frac{1368}{64624199}a^{15}-\frac{16832156}{64624199}a^{14}-\frac{17955}{64624199}a^{13}-\frac{24108677}{64624199}a^{12}+\frac{140049}{64624199}a^{11}-\frac{9541767}{64624199}a^{10}-\frac{660231}{64624199}a^{9}+\frac{14779571}{64624199}a^{8}+\frac{1828332}{64624199}a^{7}-\frac{12944266}{64624199}a^{6}-\frac{2742498}{64624199}a^{5}+\frac{10933285}{64624199}a^{4}+\frac{1869885}{64624199}a^{3}-\frac{21076028}{64624199}a^{2}-\frac{373977}{64624199}a-\frac{10406257}{64624199}$, $\frac{1}{64624199}a^{20}-\frac{60}{64624199}a^{18}-\frac{18551208}{64624199}a^{17}+\frac{1530}{64624199}a^{16}-\frac{23251377}{64624199}a^{15}-\frac{21600}{64624199}a^{14}-\frac{28714675}{64624199}a^{13}+\frac{184275}{64624199}a^{12}-\frac{12389502}{64624199}a^{11}-\frac{972972}{64624199}a^{10}+\frac{18972579}{64624199}a^{9}+\frac{3127410}{64624199}a^{8}-\frac{29526765}{64624199}a^{7}-\frac{5773680}{64624199}a^{6}+\frac{3494934}{64624199}a^{5}+\frac{5412825}{64624199}a^{4}-\frac{30691743}{64624199}a^{3}-\frac{1968300}{64624199}a^{2}-\frac{8483114}{64624199}a+\frac{118098}{64624199}$, $\frac{1}{64624199}a^{21}-\frac{1830174}{64624199}a^{18}-\frac{1890}{64624199}a^{17}-\frac{21448427}{64624199}a^{16}+\frac{60480}{64624199}a^{15}-\frac{4656851}{64624199}a^{14}-\frac{893025}{64624199}a^{13}+\frac{27446455}{64624199}a^{12}+\frac{7429968}{64624199}a^{11}+\frac{28084350}{64624199}a^{10}+\frac{28137749}{64624199}a^{9}+\frac{17132908}{64624199}a^{8}-\frac{25322158}{64624199}a^{7}+\frac{2329362}{64624199}a^{6}-\frac{29888657}{64624199}a^{5}-\frac{20936633}{64624199}a^{4}-\frac{19023598}{64624199}a^{3}+\frac{19439186}{64624199}a^{2}-\frac{22320522}{64624199}a+\frac{21866570}{64624199}$, $\frac{1}{64624199}a^{22}-\frac{2079}{64624199}a^{18}+\frac{3480053}{64624199}a^{17}+\frac{70686}{64624199}a^{16}-\frac{21322580}{64624199}a^{15}-\frac{1122660}{64624199}a^{14}-\frac{4234623}{64624199}a^{13}+\frac{10216206}{64624199}a^{12}-\frac{22074557}{64624199}a^{11}+\frac{8435066}{64624199}a^{10}+\frac{22795616}{64624199}a^{9}-\frac{8104443}{64624199}a^{8}-\frac{8380891}{64624199}a^{7}-\frac{26980526}{64624199}a^{6}+\frac{375902}{885263}a^{5}+\frac{10309025}{64624199}a^{4}-\frac{4733068}{64624199}a^{3}+\frac{6485527}{64624199}a^{2}+\frac{13776181}{64624199}a+\frac{7440174}{64624199}$, $\frac{1}{64624199}a^{23}+\frac{7708510}{64624199}a^{18}-\frac{47817}{64624199}a^{17}+\frac{8837538}{64624199}a^{16}+\frac{1721412}{64624199}a^{15}+\frac{28028911}{64624199}a^{14}-\frac{27112239}{64624199}a^{13}+\frac{4364384}{64624199}a^{12}-\frac{23524058}{64624199}a^{11}+\frac{25091116}{64624199}a^{10}-\frac{23616513}{64624199}a^{9}+\frac{21852693}{64624199}a^{8}+\frac{25918160}{64624199}a^{7}-\frac{21384}{64624199}a^{6}-\frac{4414805}{64624199}a^{5}-\frac{22151601}{64624199}a^{4}+\frac{16524502}{64624199}a^{3}+\frac{11920891}{64624199}a^{2}+\frac{5432379}{64624199}a+\frac{14498362}{64624199}$, $\frac{1}{64624199}a^{24}-\frac{54648}{64624199}a^{18}-\frac{4146785}{64624199}a^{17}+\frac{2090286}{64624199}a^{16}+\frac{16531668}{64624199}a^{15}+\frac{29212295}{64624199}a^{14}-\frac{196888}{885263}a^{13}+\frac{12554345}{64624199}a^{12}+\frac{3218421}{64624199}a^{11}-\frac{24861581}{64624199}a^{10}+\frac{4950457}{64624199}a^{9}+\frac{11205612}{64624199}a^{8}-\frac{17839391}{64624199}a^{7}+\frac{8373074}{64624199}a^{6}-\frac{27736690}{64624199}a^{5}+\frac{5774007}{64624199}a^{4}+\frac{24541297}{64624199}a^{3}-\frac{5885948}{64624199}a^{2}-\frac{950559}{64624199}a+\frac{10412350}{64624199}$, $\frac{1}{64624199}a^{25}-\frac{13015142}{64624199}a^{18}-\frac{1024650}{64624199}a^{17}-\frac{21570646}{64624199}a^{16}-\frac{25277639}{64624199}a^{15}+\frac{2814654}{64624199}a^{14}+\frac{712490}{64624199}a^{13}+\frac{5782738}{64624199}a^{12}+\frac{2880689}{64624199}a^{11}+\frac{19129172}{64624199}a^{10}-\frac{8795034}{64624199}a^{9}-\frac{17082485}{64624199}a^{8}+\frac{14048556}{64624199}a^{7}-\frac{404148}{885263}a^{6}-\frac{2739216}{64624199}a^{5}-\frac{8643977}{64624199}a^{4}+\frac{8730913}{64624199}a^{3}-\frac{31254125}{64624199}a^{2}-\frac{5435862}{64624199}a+\frac{11818664}{64624199}$, $\frac{1}{64624199}a^{26}-\frac{1210950}{64624199}a^{18}+\frac{12056648}{64624199}a^{17}-\frac{15217439}{64624199}a^{16}-\frac{28750014}{64624199}a^{15}-\frac{31769413}{64624199}a^{14}+\frac{11712}{64624199}a^{13}-\frac{29525268}{64624199}a^{12}-\frac{13405864}{64624199}a^{11}+\frac{21872765}{64624199}a^{10}-\frac{2183456}{64624199}a^{9}-\frac{28832195}{64624199}a^{8}-\frac{16079639}{64624199}a^{7}+\frac{11150271}{64624199}a^{6}+\frac{3533375}{64624199}a^{5}-\frac{20092682}{64624199}a^{4}+\frac{25067334}{64624199}a^{3}-\frac{8621294}{64624199}a^{2}+\frac{13479212}{64624199}a+\frac{31351313}{64624199}$, $\frac{1}{64624199}a^{27}+\frac{16758175}{64624199}a^{18}-\frac{19617390}{64624199}a^{17}-\frac{24135676}{64624199}a^{16}+\frac{9205212}{64624199}a^{15}+\frac{25437505}{64624199}a^{14}+\frac{6222545}{64624199}a^{13}+\frac{18448629}{64624199}a^{12}-\frac{24313060}{64624199}a^{11}+\frac{7976497}{64624199}a^{10}-\frac{4971617}{64624199}a^{9}+\frac{21254955}{64624199}a^{8}+\frac{4727931}{64624199}a^{7}+\frac{2584921}{64624199}a^{6}-\frac{10459172}{64624199}a^{5}-\frac{2359444}{64624199}a^{4}+\frac{25934894}{64624199}a^{3}+\frac{32283682}{64624199}a^{2}-\frac{14334444}{64624199}a+\frac{3394054}{64624199}$, $\frac{1}{64624199}a^{28}-\frac{23882040}{64624199}a^{18}+\frac{26341513}{64624199}a^{17}-\frac{19000484}{64624199}a^{16}-\frac{22779449}{64624199}a^{15}-\frac{5391285}{64624199}a^{14}+\frac{21210210}{64624199}a^{13}+\frac{30435812}{64624199}a^{12}-\frac{638995}{64624199}a^{11}-\frac{32075246}{64624199}a^{10}-\frac{21217410}{64624199}a^{9}+\frac{7948207}{64624199}a^{8}-\frac{9047697}{64624199}a^{7}+\frac{25109650}{64624199}a^{6}+\frac{15089483}{64624199}a^{5}+\frac{5342829}{64624199}a^{4}-\frac{6050486}{64624199}a^{3}+\frac{11088035}{64624199}a^{2}-\frac{14788792}{64624199}a-\frac{11457102}{64624199}$, $\frac{1}{64624199}a^{29}-\frac{15242142}{64624199}a^{18}-\frac{23168585}{64624199}a^{17}+\frac{25515155}{64624199}a^{16}+\frac{30018940}{64624199}a^{15}-\frac{31431196}{64624199}a^{14}+\frac{9967977}{64624199}a^{13}+\frac{451301}{64624199}a^{12}-\frac{1674531}{64624199}a^{11}-\frac{1231275}{64624199}a^{10}+\frac{3110977}{64624199}a^{9}-\frac{27442828}{64624199}a^{8}+\frac{13649595}{64624199}a^{7}+\frac{14928656}{64624199}a^{6}-\frac{9779188}{64624199}a^{5}-\frac{20969462}{64624199}a^{4}+\frac{836256}{64624199}a^{3}+\frac{4849587}{64624199}a^{2}-\frac{22331586}{64624199}a-\frac{13422139}{64624199}$, $\frac{1}{64624199}a^{30}+\frac{19507418}{64624199}a^{18}-\frac{3172352}{64624199}a^{17}-\frac{12638257}{64624199}a^{16}+\frac{10826982}{64624199}a^{15}-\frac{31920175}{64624199}a^{14}+\frac{11274456}{64624199}a^{13}-\frac{14733492}{64624199}a^{12}-\frac{21027685}{64624199}a^{11}-\frac{10711840}{64624199}a^{10}-\frac{235687}{885263}a^{9}+\frac{1472159}{64624199}a^{8}+\frac{11433627}{64624199}a^{7}-\frac{24447970}{64624199}a^{6}+\frac{16585181}{64624199}a^{5}+\frac{2874630}{64624199}a^{4}-\frac{23693315}{64624199}a^{3}+\frac{21988085}{64624199}a^{2}+\frac{18136121}{64624199}a-\frac{12856894}{64624199}$, $\frac{1}{64624199}a^{31}+\frac{26692314}{64624199}a^{18}+\frac{673186}{64624199}a^{17}+\frac{13739993}{64624199}a^{16}-\frac{28273812}{64624199}a^{15}-\frac{21956192}{64624199}a^{14}-\frac{22201882}{64624199}a^{13}-\frac{1266070}{64624199}a^{12}-\frac{17082597}{64624199}a^{11}+\frac{13844526}{64624199}a^{10}-\frac{5422386}{64624199}a^{9}-\frac{23091202}{64624199}a^{8}-\frac{30210845}{64624199}a^{7}-\frac{10563883}{64624199}a^{6}-\frac{20793157}{64624199}a^{5}+\frac{29021240}{64624199}a^{4}+\frac{5813113}{64624199}a^{3}+\frac{10570621}{64624199}a^{2}+\frac{16227780}{64624199}a-\frac{9013548}{64624199}$, $\frac{1}{64624199}a^{32}+\frac{6031614}{64624199}a^{18}-\frac{15778885}{64624199}a^{17}+\frac{5492071}{64624199}a^{16}-\frac{24369309}{64624199}a^{15}+\frac{25868238}{64624199}a^{14}+\frac{6172016}{64624199}a^{13}+\frac{24367204}{64624199}a^{12}-\frac{31247705}{64624199}a^{11}-\frac{16954826}{64624199}a^{10}-\frac{13618167}{64624199}a^{9}+\frac{27635321}{64624199}a^{8}+\frac{27827296}{64624199}a^{7}+\frac{16563653}{64624199}a^{6}+\frac{3746571}{64624199}a^{5}+\frac{4952350}{64624199}a^{4}-\frac{16258604}{64624199}a^{3}-\frac{25390606}{64624199}a^{2}-\frac{3647703}{64624199}a-\frac{6491112}{64624199}$, $\frac{1}{64624199}a^{33}-\frac{28226030}{64624199}a^{18}+\frac{26173074}{64624199}a^{17}+\frac{1534531}{64624199}a^{16}-\frac{18106441}{64624199}a^{15}+\frac{4953407}{64624199}a^{14}+\frac{11839050}{64624199}a^{13}-\frac{3537076}{64624199}a^{12}+\frac{29065416}{64624199}a^{11}-\frac{1851261}{64624199}a^{10}+\frac{13787377}{64624199}a^{9}-\frac{16079529}{64624199}a^{8}+\frac{20114160}{64624199}a^{7}-\frac{1511169}{64624199}a^{6}+\frac{31938689}{64624199}a^{5}+\frac{4993762}{64624199}a^{4}+\frac{23771280}{64624199}a^{3}+\frac{6175992}{64624199}a^{2}-\frac{29248329}{64624199}a+\frac{13633252}{64624199}$, $\frac{1}{64624199}a^{34}-\frac{20455687}{64624199}a^{18}+\frac{8255796}{64624199}a^{17}-\frac{20497108}{64624199}a^{16}-\frac{27108555}{64624199}a^{15}-\frac{31307241}{64624199}a^{14}-\frac{18937168}{64624199}a^{13}+\frac{22655315}{64624199}a^{12}+\frac{27795578}{64624199}a^{11}+\frac{8339792}{64624199}a^{10}+\frac{28797370}{64624199}a^{9}-\frac{21881609}{64624199}a^{8}-\frac{6479445}{64624199}a^{7}-\frac{20860981}{64624199}a^{6}+\frac{12446176}{64624199}a^{5}+\frac{20860981}{64624199}a^{4}+\frac{12844655}{64624199}a^{3}+\frac{7604615}{64624199}a^{2}-\frac{26475000}{64624199}a+\frac{2057130}{64624199}$, $\frac{1}{64624199}a^{35}+\frac{3807008}{64624199}a^{18}-\frac{23235685}{64624199}a^{17}+\frac{19253400}{64624199}a^{16}-\frac{30205592}{64624199}a^{15}+\frac{4218128}{64624199}a^{14}+\frac{118147}{64624199}a^{13}+\frac{18266289}{64624199}a^{12}+\frac{16106785}{64624199}a^{11}-\frac{753242}{64624199}a^{10}-\frac{12337291}{64624199}a^{9}-\frac{12979147}{64624199}a^{8}-\frac{2551570}{64624199}a^{7}-\frac{15470652}{64624199}a^{6}+\frac{25708964}{64624199}a^{5}-\frac{4881805}{64624199}a^{4}+\frac{18986490}{64624199}a^{3}-\frac{18498093}{64624199}a^{2}-\frac{219245}{64624199}a-\frac{590882}{64624199}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, 1/64624199*a^19 - 6183736/64624199*a^18 - 57/64624199*a^17 + 10800749/64624199*a^16 + 1368/64624199*a^15 - 16832156/64624199*a^14 - 17955/64624199*a^13 - 24108677/64624199*a^12 + 140049/64624199*a^11 - 9541767/64624199*a^10 - 660231/64624199*a^9 + 14779571/64624199*a^8 + 1828332/64624199*a^7 - 12944266/64624199*a^6 - 2742498/64624199*a^5 + 10933285/64624199*a^4 + 1869885/64624199*a^3 - 21076028/64624199*a^2 - 373977/64624199*a - 10406257/64624199, 1/64624199*a^20 - 60/64624199*a^18 - 18551208/64624199*a^17 + 1530/64624199*a^16 - 23251377/64624199*a^15 - 21600/64624199*a^14 - 28714675/64624199*a^13 + 184275/64624199*a^12 - 12389502/64624199*a^11 - 972972/64624199*a^10 + 18972579/64624199*a^9 + 3127410/64624199*a^8 - 29526765/64624199*a^7 - 5773680/64624199*a^6 + 3494934/64624199*a^5 + 5412825/64624199*a^4 - 30691743/64624199*a^3 - 1968300/64624199*a^2 - 8483114/64624199*a + 118098/64624199, 1/64624199*a^21 - 1830174/64624199*a^18 - 1890/64624199*a^17 - 21448427/64624199*a^16 + 60480/64624199*a^15 - 4656851/64624199*a^14 - 893025/64624199*a^13 + 27446455/64624199*a^12 + 7429968/64624199*a^11 + 28084350/64624199*a^10 + 28137749/64624199*a^9 + 17132908/64624199*a^8 - 25322158/64624199*a^7 + 2329362/64624199*a^6 - 29888657/64624199*a^5 - 20936633/64624199*a^4 - 19023598/64624199*a^3 + 19439186/64624199*a^2 - 22320522/64624199*a + 21866570/64624199, 1/64624199*a^22 - 2079/64624199*a^18 + 3480053/64624199*a^17 + 70686/64624199*a^16 - 21322580/64624199*a^15 - 1122660/64624199*a^14 - 4234623/64624199*a^13 + 10216206/64624199*a^12 - 22074557/64624199*a^11 + 8435066/64624199*a^10 + 22795616/64624199*a^9 - 8104443/64624199*a^8 - 8380891/64624199*a^7 - 26980526/64624199*a^6 + 375902/885263*a^5 + 10309025/64624199*a^4 - 4733068/64624199*a^3 + 6485527/64624199*a^2 + 13776181/64624199*a + 7440174/64624199, 1/64624199*a^23 + 7708510/64624199*a^18 - 47817/64624199*a^17 + 8837538/64624199*a^16 + 1721412/64624199*a^15 + 28028911/64624199*a^14 - 27112239/64624199*a^13 + 4364384/64624199*a^12 - 23524058/64624199*a^11 + 25091116/64624199*a^10 - 23616513/64624199*a^9 + 21852693/64624199*a^8 + 25918160/64624199*a^7 - 21384/64624199*a^6 - 4414805/64624199*a^5 - 22151601/64624199*a^4 + 16524502/64624199*a^3 + 11920891/64624199*a^2 + 5432379/64624199*a + 14498362/64624199, 1/64624199*a^24 - 54648/64624199*a^18 - 4146785/64624199*a^17 + 2090286/64624199*a^16 + 16531668/64624199*a^15 + 29212295/64624199*a^14 - 196888/885263*a^13 + 12554345/64624199*a^12 + 3218421/64624199*a^11 - 24861581/64624199*a^10 + 4950457/64624199*a^9 + 11205612/64624199*a^8 - 17839391/64624199*a^7 + 8373074/64624199*a^6 - 27736690/64624199*a^5 + 5774007/64624199*a^4 + 24541297/64624199*a^3 - 5885948/64624199*a^2 - 950559/64624199*a + 10412350/64624199, 1/64624199*a^25 - 13015142/64624199*a^18 - 1024650/64624199*a^17 - 21570646/64624199*a^16 - 25277639/64624199*a^15 + 2814654/64624199*a^14 + 712490/64624199*a^13 + 5782738/64624199*a^12 + 2880689/64624199*a^11 + 19129172/64624199*a^10 - 8795034/64624199*a^9 - 17082485/64624199*a^8 + 14048556/64624199*a^7 - 404148/885263*a^6 - 2739216/64624199*a^5 - 8643977/64624199*a^4 + 8730913/64624199*a^3 - 31254125/64624199*a^2 - 5435862/64624199*a + 11818664/64624199, 1/64624199*a^26 - 1210950/64624199*a^18 + 12056648/64624199*a^17 - 15217439/64624199*a^16 - 28750014/64624199*a^15 - 31769413/64624199*a^14 + 11712/64624199*a^13 - 29525268/64624199*a^12 - 13405864/64624199*a^11 + 21872765/64624199*a^10 - 2183456/64624199*a^9 - 28832195/64624199*a^8 - 16079639/64624199*a^7 + 11150271/64624199*a^6 + 3533375/64624199*a^5 - 20092682/64624199*a^4 + 25067334/64624199*a^3 - 8621294/64624199*a^2 + 13479212/64624199*a + 31351313/64624199, 1/64624199*a^27 + 16758175/64624199*a^18 - 19617390/64624199*a^17 - 24135676/64624199*a^16 + 9205212/64624199*a^15 + 25437505/64624199*a^14 + 6222545/64624199*a^13 + 18448629/64624199*a^12 - 24313060/64624199*a^11 + 7976497/64624199*a^10 - 4971617/64624199*a^9 + 21254955/64624199*a^8 + 4727931/64624199*a^7 + 2584921/64624199*a^6 - 10459172/64624199*a^5 - 2359444/64624199*a^4 + 25934894/64624199*a^3 + 32283682/64624199*a^2 - 14334444/64624199*a + 3394054/64624199, 1/64624199*a^28 - 23882040/64624199*a^18 + 26341513/64624199*a^17 - 19000484/64624199*a^16 - 22779449/64624199*a^15 - 5391285/64624199*a^14 + 21210210/64624199*a^13 + 30435812/64624199*a^12 - 638995/64624199*a^11 - 32075246/64624199*a^10 - 21217410/64624199*a^9 + 7948207/64624199*a^8 - 9047697/64624199*a^7 + 25109650/64624199*a^6 + 15089483/64624199*a^5 + 5342829/64624199*a^4 - 6050486/64624199*a^3 + 11088035/64624199*a^2 - 14788792/64624199*a - 11457102/64624199, 1/64624199*a^29 - 15242142/64624199*a^18 - 23168585/64624199*a^17 + 25515155/64624199*a^16 + 30018940/64624199*a^15 - 31431196/64624199*a^14 + 9967977/64624199*a^13 + 451301/64624199*a^12 - 1674531/64624199*a^11 - 1231275/64624199*a^10 + 3110977/64624199*a^9 - 27442828/64624199*a^8 + 13649595/64624199*a^7 + 14928656/64624199*a^6 - 9779188/64624199*a^5 - 20969462/64624199*a^4 + 836256/64624199*a^3 + 4849587/64624199*a^2 - 22331586/64624199*a - 13422139/64624199, 1/64624199*a^30 + 19507418/64624199*a^18 - 3172352/64624199*a^17 - 12638257/64624199*a^16 + 10826982/64624199*a^15 - 31920175/64624199*a^14 + 11274456/64624199*a^13 - 14733492/64624199*a^12 - 21027685/64624199*a^11 - 10711840/64624199*a^10 - 235687/885263*a^9 + 1472159/64624199*a^8 + 11433627/64624199*a^7 - 24447970/64624199*a^6 + 16585181/64624199*a^5 + 2874630/64624199*a^4 - 23693315/64624199*a^3 + 21988085/64624199*a^2 + 18136121/64624199*a - 12856894/64624199, 1/64624199*a^31 + 26692314/64624199*a^18 + 673186/64624199*a^17 + 13739993/64624199*a^16 - 28273812/64624199*a^15 - 21956192/64624199*a^14 - 22201882/64624199*a^13 - 1266070/64624199*a^12 - 17082597/64624199*a^11 + 13844526/64624199*a^10 - 5422386/64624199*a^9 - 23091202/64624199*a^8 - 30210845/64624199*a^7 - 10563883/64624199*a^6 - 20793157/64624199*a^5 + 29021240/64624199*a^4 + 5813113/64624199*a^3 + 10570621/64624199*a^2 + 16227780/64624199*a - 9013548/64624199, 1/64624199*a^32 + 6031614/64624199*a^18 - 15778885/64624199*a^17 + 5492071/64624199*a^16 - 24369309/64624199*a^15 + 25868238/64624199*a^14 + 6172016/64624199*a^13 + 24367204/64624199*a^12 - 31247705/64624199*a^11 - 16954826/64624199*a^10 - 13618167/64624199*a^9 + 27635321/64624199*a^8 + 27827296/64624199*a^7 + 16563653/64624199*a^6 + 3746571/64624199*a^5 + 4952350/64624199*a^4 - 16258604/64624199*a^3 - 25390606/64624199*a^2 - 3647703/64624199*a - 6491112/64624199, 1/64624199*a^33 - 28226030/64624199*a^18 + 26173074/64624199*a^17 + 1534531/64624199*a^16 - 18106441/64624199*a^15 + 4953407/64624199*a^14 + 11839050/64624199*a^13 - 3537076/64624199*a^12 + 29065416/64624199*a^11 - 1851261/64624199*a^10 + 13787377/64624199*a^9 - 16079529/64624199*a^8 + 20114160/64624199*a^7 - 1511169/64624199*a^6 + 31938689/64624199*a^5 + 4993762/64624199*a^4 + 23771280/64624199*a^3 + 6175992/64624199*a^2 - 29248329/64624199*a + 13633252/64624199, 1/64624199*a^34 - 20455687/64624199*a^18 + 8255796/64624199*a^17 - 20497108/64624199*a^16 - 27108555/64624199*a^15 - 31307241/64624199*a^14 - 18937168/64624199*a^13 + 22655315/64624199*a^12 + 27795578/64624199*a^11 + 8339792/64624199*a^10 + 28797370/64624199*a^9 - 21881609/64624199*a^8 - 6479445/64624199*a^7 - 20860981/64624199*a^6 + 12446176/64624199*a^5 + 20860981/64624199*a^4 + 12844655/64624199*a^3 + 7604615/64624199*a^2 - 26475000/64624199*a + 2057130/64624199, 1/64624199*a^35 + 3807008/64624199*a^18 - 23235685/64624199*a^17 + 19253400/64624199*a^16 - 30205592/64624199*a^15 + 4218128/64624199*a^14 + 118147/64624199*a^13 + 18266289/64624199*a^12 + 16106785/64624199*a^11 - 753242/64624199*a^10 - 12337291/64624199*a^9 - 12979147/64624199*a^8 - 2551570/64624199*a^7 - 15470652/64624199*a^6 + 25708964/64624199*a^5 - 4881805/64624199*a^4 + 18986490/64624199*a^3 - 18498093/64624199*a^2 - 219245/64624199*a - 590882/64624199

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$35$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	not computed	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 - 110*x^34 + 110*x^33 + 5551*x^32 - 5551*x^31 - 170273*x^30 + 170273*x^29 + 3546007*x^28 - 3546007*x^27 - 53034356*x^26 + 53034356*x^25 + 587601367*x^24 - 587601367*x^23 - 4903561973*x^22 + 4903561973*x^21 + 31025687812*x^20 - 31025687812*x^19 - 148620561113*x^18 + 148620561113*x^17 + 534035184802*x^16 - 534035184802*x^15 - 1413681908438*x^14 + 1413681908438*x^13 + 2676523987366*x^12 - 2676523987366*x^11 - 3458784856340*x^10 + 3458784856340*x^9 + 2828954020750*x^8 - 2828954020750*x^7 - 1286656880618*x^6 + 1286656880618*x^5 + 256697207395*x^4 - 256697207395*x^3 - 15659396372*x^2 + 15659396372*x - 1324838279);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_{36}$ (as 36T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A cyclic group of order 36

The 36 conjugacy class representatives for $C_{36}$

Character table for $C_{36}$ is not computed

Intermediate fields

$\Q(\sqrt{37}) $, 3.3.1369.1, 4.4.6129013.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.315191919957242668714693.1, $\Q(\zeta_{37})^+$

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$36$	$18^{2}$	$36$	$18^{2}$	R	$36$	$36$	$36$	${\href{/padicField/23.12.0.1}{12} }^{3}$	${\href{/padicField/29.12.0.1}{12} }^{3}$	${\href{/padicField/31.4.0.1}{4} }^{9}$	R	${\href{/padicField/41.9.0.1}{9} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{9}$	${\href{/padicField/47.3.0.1}{3} }^{12}$	${\href{/padicField/53.9.0.1}{9} }^{4}$	$36$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$11$ 11	11.6.3.1	$x^{6} + 242 x^{2} - 11979$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.1	$x^{6} + 242 x^{2} - 11979$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.1	$x^{6} + 242 x^{2} - 11979$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.1	$x^{6} + 242 x^{2} - 11979$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.1	$x^{6} + 242 x^{2} - 11979$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	11.6.3.1	$x^{6} + 242 x^{2} - 11979$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$37$ 37		Deg $36$	$36$	$1$	$35$

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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