Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512)
gp: K = bnfinit(y^36 + 68*y^34 + 2018*y^32 + 34520*y^30 + 379376*y^28 + 2832128*y^26 + 14836664*y^24 + 55646624*y^22 + 151170256*y^20 + 298819648*y^18 + 428794592*y^16 + 442226304*y^14 + 321513984*y^12 + 159622144*y^10 + 51473664*y^8 + 9968640*y^6 + 1025280*y^4 + 46080*y^2 + 512, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512)
\( x^{36} + 68 x^{34} + 2018 x^{32} + 34520 x^{30} + 379376 x^{28} + 2832128 x^{26} + 14836664 x^{24} + \cdots + 512 \)
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: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(52733281945045886724167383478270850720626086921526306402773390818541568\)
\(\medspace = 2^{99}\cdot 19^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(92.15\) | 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: | $2^{11/4}19^{8/9}\approx 92.151488696787$ | ||
| Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(304=2^{4}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{304}(1,·)$, $\chi_{304}(131,·)$, $\chi_{304}(9,·)$, $\chi_{304}(11,·)$, $\chi_{304}(17,·)$, $\chi_{304}(275,·)$, $\chi_{304}(73,·)$, $\chi_{304}(25,·)$, $\chi_{304}(153,·)$, $\chi_{304}(283,·)$, $\chi_{304}(289,·)$, $\chi_{304}(35,·)$, $\chi_{304}(49,·)$, $\chi_{304}(169,·)$, $\chi_{304}(43,·)$, $\chi_{304}(177,·)$, $\chi_{304}(115,·)$, $\chi_{304}(137,·)$, $\chi_{304}(187,·)$, $\chi_{304}(123,·)$, $\chi_{304}(267,·)$, $\chi_{304}(161,·)$, $\chi_{304}(201,·)$, $\chi_{304}(291,·)$, $\chi_{304}(81,·)$, $\chi_{304}(163,·)$, $\chi_{304}(139,·)$, $\chi_{304}(225,·)$, $\chi_{304}(99,·)$, $\chi_{304}(195,·)$, $\chi_{304}(273,·)$, $\chi_{304}(233,·)$, $\chi_{304}(235,·)$, $\chi_{304}(83,·)$, $\chi_{304}(121,·)$, $\chi_{304}(251,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{32}a^{22}$, $\frac{1}{32}a^{23}$, $\frac{1}{64}a^{24}$, $\frac{1}{64}a^{25}$, $\frac{1}{64}a^{26}$, $\frac{1}{64}a^{27}$, $\frac{1}{128}a^{28}$, $\frac{1}{128}a^{29}$, $\frac{1}{128}a^{30}$, $\frac{1}{128}a^{31}$, $\frac{1}{256}a^{32}$, $\frac{1}{256}a^{33}$, $\frac{1}{98\!\cdots\!96}a^{34}-\frac{30\!\cdots\!55}{98\!\cdots\!96}a^{32}-\frac{78\!\cdots\!55}{49\!\cdots\!48}a^{30}-\frac{55\!\cdots\!83}{30\!\cdots\!28}a^{28}-\frac{30\!\cdots\!93}{12\!\cdots\!12}a^{26}-\frac{16\!\cdots\!75}{24\!\cdots\!24}a^{24}+\frac{13\!\cdots\!61}{12\!\cdots\!12}a^{22}-\frac{48\!\cdots\!07}{12\!\cdots\!12}a^{20}-\frac{24\!\cdots\!37}{15\!\cdots\!64}a^{18}+\frac{89\!\cdots\!68}{38\!\cdots\!41}a^{16}+\frac{10\!\cdots\!25}{30\!\cdots\!28}a^{14}+\frac{43\!\cdots\!96}{38\!\cdots\!41}a^{12}-\frac{10\!\cdots\!83}{15\!\cdots\!64}a^{10}-\frac{16\!\cdots\!41}{15\!\cdots\!64}a^{8}+\frac{11\!\cdots\!97}{76\!\cdots\!82}a^{6}-\frac{74\!\cdots\!08}{38\!\cdots\!41}a^{4}+\frac{12\!\cdots\!05}{38\!\cdots\!41}a^{2}-\frac{10\!\cdots\!26}{38\!\cdots\!41}$, $\frac{1}{98\!\cdots\!96}a^{35}-\frac{30\!\cdots\!55}{98\!\cdots\!96}a^{33}-\frac{78\!\cdots\!55}{49\!\cdots\!48}a^{31}-\frac{55\!\cdots\!83}{30\!\cdots\!28}a^{29}-\frac{30\!\cdots\!93}{12\!\cdots\!12}a^{27}-\frac{16\!\cdots\!75}{24\!\cdots\!24}a^{25}+\frac{13\!\cdots\!61}{12\!\cdots\!12}a^{23}-\frac{48\!\cdots\!07}{12\!\cdots\!12}a^{21}-\frac{24\!\cdots\!37}{15\!\cdots\!64}a^{19}+\frac{89\!\cdots\!68}{38\!\cdots\!41}a^{17}+\frac{10\!\cdots\!25}{30\!\cdots\!28}a^{15}+\frac{43\!\cdots\!96}{38\!\cdots\!41}a^{13}-\frac{10\!\cdots\!83}{15\!\cdots\!64}a^{11}-\frac{16\!\cdots\!41}{15\!\cdots\!64}a^{9}+\frac{11\!\cdots\!97}{76\!\cdots\!82}a^{7}-\frac{74\!\cdots\!08}{38\!\cdots\!41}a^{5}+\frac{12\!\cdots\!05}{38\!\cdots\!41}a^{3}-\frac{10\!\cdots\!26}{38\!\cdots\!41}a$
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
$C_{37}\times C_{294409}$, which has order $10893133$ (assuming GRH)
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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$\frac{58\!\cdots\!59}{30\!\cdots\!28}a^{34}+\frac{12\!\cdots\!17}{98\!\cdots\!96}a^{32}+\frac{13\!\cdots\!19}{38\!\cdots\!41}a^{30}+\frac{27\!\cdots\!11}{49\!\cdots\!48}a^{28}+\frac{70\!\cdots\!09}{12\!\cdots\!12}a^{26}+\frac{47\!\cdots\!37}{12\!\cdots\!12}a^{24}+\frac{10\!\cdots\!05}{61\!\cdots\!56}a^{22}+\frac{69\!\cdots\!67}{12\!\cdots\!12}a^{20}+\frac{38\!\cdots\!85}{30\!\cdots\!28}a^{18}+\frac{11\!\cdots\!91}{61\!\cdots\!56}a^{16}+\frac{70\!\cdots\!07}{38\!\cdots\!41}a^{14}+\frac{33\!\cdots\!57}{30\!\cdots\!28}a^{12}+\frac{23\!\cdots\!95}{76\!\cdots\!82}a^{10}-\frac{25\!\cdots\!19}{76\!\cdots\!82}a^{8}-\frac{16\!\cdots\!00}{38\!\cdots\!41}a^{6}-\frac{33\!\cdots\!49}{38\!\cdots\!41}a^{4}-\frac{21\!\cdots\!96}{38\!\cdots\!41}a^{2}-\frac{46\!\cdots\!27}{38\!\cdots\!41}$, $\frac{59\!\cdots\!55}{38\!\cdots\!41}a^{34}+\frac{32\!\cdots\!39}{30\!\cdots\!28}a^{32}+\frac{37\!\cdots\!35}{12\!\cdots\!12}a^{30}+\frac{31\!\cdots\!15}{61\!\cdots\!56}a^{28}+\frac{33\!\cdots\!43}{61\!\cdots\!56}a^{26}+\frac{11\!\cdots\!35}{30\!\cdots\!28}a^{24}+\frac{28\!\cdots\!27}{15\!\cdots\!64}a^{22}+\frac{45\!\cdots\!49}{76\!\cdots\!82}a^{20}+\frac{19\!\cdots\!91}{15\!\cdots\!64}a^{18}+\frac{11\!\cdots\!95}{76\!\cdots\!82}a^{16}+\frac{15\!\cdots\!83}{38\!\cdots\!41}a^{14}-\frac{68\!\cdots\!38}{38\!\cdots\!41}a^{12}-\frac{11\!\cdots\!70}{38\!\cdots\!41}a^{10}-\frac{37\!\cdots\!95}{15\!\cdots\!64}a^{8}-\frac{38\!\cdots\!16}{38\!\cdots\!41}a^{6}-\frac{77\!\cdots\!02}{38\!\cdots\!41}a^{4}-\frac{52\!\cdots\!96}{38\!\cdots\!41}a^{2}-\frac{43\!\cdots\!58}{38\!\cdots\!41}$, $\frac{40\!\cdots\!25}{61\!\cdots\!56}a^{34}+\frac{26\!\cdots\!39}{61\!\cdots\!56}a^{32}+\frac{37\!\cdots\!43}{30\!\cdots\!28}a^{30}+\frac{59\!\cdots\!91}{30\!\cdots\!28}a^{28}+\frac{15\!\cdots\!61}{76\!\cdots\!82}a^{26}+\frac{50\!\cdots\!95}{38\!\cdots\!41}a^{24}+\frac{44\!\cdots\!71}{76\!\cdots\!82}a^{22}+\frac{21\!\cdots\!65}{12\!\cdots\!12}a^{20}+\frac{11\!\cdots\!85}{30\!\cdots\!28}a^{18}+\frac{14\!\cdots\!85}{30\!\cdots\!28}a^{16}+\frac{55\!\cdots\!61}{15\!\cdots\!64}a^{14}+\frac{18\!\cdots\!19}{30\!\cdots\!28}a^{12}-\frac{11\!\cdots\!93}{76\!\cdots\!82}a^{10}-\frac{10\!\cdots\!59}{76\!\cdots\!82}a^{8}-\frac{20\!\cdots\!06}{38\!\cdots\!41}a^{6}-\frac{69\!\cdots\!87}{76\!\cdots\!82}a^{4}-\frac{21\!\cdots\!86}{38\!\cdots\!41}a^{2}-\frac{58\!\cdots\!39}{38\!\cdots\!41}$, $\frac{50\!\cdots\!33}{61\!\cdots\!56}a^{34}+\frac{33\!\cdots\!19}{61\!\cdots\!56}a^{32}+\frac{49\!\cdots\!79}{30\!\cdots\!28}a^{30}+\frac{83\!\cdots\!31}{30\!\cdots\!28}a^{28}+\frac{22\!\cdots\!89}{76\!\cdots\!82}a^{26}+\frac{81\!\cdots\!63}{38\!\cdots\!41}a^{24}+\frac{16\!\cdots\!79}{15\!\cdots\!64}a^{22}+\frac{47\!\cdots\!73}{12\!\cdots\!12}a^{20}+\frac{30\!\cdots\!61}{30\!\cdots\!28}a^{18}+\frac{11\!\cdots\!17}{61\!\cdots\!56}a^{16}+\frac{37\!\cdots\!81}{15\!\cdots\!64}a^{14}+\frac{71\!\cdots\!99}{30\!\cdots\!28}a^{12}+\frac{11\!\cdots\!31}{76\!\cdots\!82}a^{10}+\frac{50\!\cdots\!63}{76\!\cdots\!82}a^{8}+\frac{66\!\cdots\!02}{38\!\cdots\!41}a^{6}+\frac{18\!\cdots\!81}{76\!\cdots\!82}a^{4}+\frac{52\!\cdots\!90}{38\!\cdots\!41}a^{2}+\frac{50\!\cdots\!96}{38\!\cdots\!41}$, $\frac{77\!\cdots\!03}{49\!\cdots\!48}a^{34}+\frac{20\!\cdots\!87}{98\!\cdots\!96}a^{32}+\frac{23\!\cdots\!91}{24\!\cdots\!24}a^{30}+\frac{11\!\cdots\!53}{49\!\cdots\!48}a^{28}+\frac{21\!\cdots\!71}{61\!\cdots\!56}a^{26}+\frac{82\!\cdots\!57}{24\!\cdots\!24}a^{24}+\frac{13\!\cdots\!81}{61\!\cdots\!56}a^{22}+\frac{58\!\cdots\!63}{61\!\cdots\!56}a^{20}+\frac{89\!\cdots\!03}{30\!\cdots\!28}a^{18}+\frac{38\!\cdots\!37}{61\!\cdots\!56}a^{16}+\frac{13\!\cdots\!97}{15\!\cdots\!64}a^{14}+\frac{13\!\cdots\!23}{15\!\cdots\!64}a^{12}+\frac{22\!\cdots\!16}{38\!\cdots\!41}a^{10}+\frac{18\!\cdots\!19}{76\!\cdots\!82}a^{8}+\frac{22\!\cdots\!86}{38\!\cdots\!41}a^{6}+\frac{28\!\cdots\!18}{38\!\cdots\!41}a^{4}+\frac{14\!\cdots\!80}{38\!\cdots\!41}a^{2}-\frac{58\!\cdots\!85}{38\!\cdots\!41}$, $\frac{12\!\cdots\!01}{76\!\cdots\!82}a^{34}+\frac{18\!\cdots\!45}{15\!\cdots\!64}a^{32}+\frac{15\!\cdots\!92}{38\!\cdots\!41}a^{30}+\frac{58\!\cdots\!85}{76\!\cdots\!82}a^{28}+\frac{36\!\cdots\!14}{38\!\cdots\!41}a^{26}+\frac{31\!\cdots\!68}{38\!\cdots\!41}a^{24}+\frac{74\!\cdots\!37}{15\!\cdots\!64}a^{22}+\frac{15\!\cdots\!13}{76\!\cdots\!82}a^{20}+\frac{47\!\cdots\!69}{76\!\cdots\!82}a^{18}+\frac{83\!\cdots\!47}{61\!\cdots\!56}a^{16}+\frac{80\!\cdots\!30}{38\!\cdots\!41}a^{14}+\frac{87\!\cdots\!85}{38\!\cdots\!41}a^{12}+\frac{64\!\cdots\!62}{38\!\cdots\!41}a^{10}+\frac{30\!\cdots\!11}{38\!\cdots\!41}a^{8}+\frac{86\!\cdots\!08}{38\!\cdots\!41}a^{6}+\frac{12\!\cdots\!84}{38\!\cdots\!41}a^{4}+\frac{74\!\cdots\!76}{38\!\cdots\!41}a^{2}+\frac{70\!\cdots\!94}{38\!\cdots\!41}$, $\frac{59\!\cdots\!55}{38\!\cdots\!41}a^{34}+\frac{32\!\cdots\!39}{30\!\cdots\!28}a^{32}+\frac{37\!\cdots\!35}{12\!\cdots\!12}a^{30}+\frac{31\!\cdots\!15}{61\!\cdots\!56}a^{28}+\frac{33\!\cdots\!43}{61\!\cdots\!56}a^{26}+\frac{11\!\cdots\!35}{30\!\cdots\!28}a^{24}+\frac{28\!\cdots\!27}{15\!\cdots\!64}a^{22}+\frac{45\!\cdots\!49}{76\!\cdots\!82}a^{20}+\frac{19\!\cdots\!91}{15\!\cdots\!64}a^{18}+\frac{11\!\cdots\!95}{76\!\cdots\!82}a^{16}+\frac{15\!\cdots\!83}{38\!\cdots\!41}a^{14}-\frac{68\!\cdots\!38}{38\!\cdots\!41}a^{12}-\frac{11\!\cdots\!70}{38\!\cdots\!41}a^{10}-\frac{37\!\cdots\!95}{15\!\cdots\!64}a^{8}-\frac{38\!\cdots\!16}{38\!\cdots\!41}a^{6}-\frac{77\!\cdots\!02}{38\!\cdots\!41}a^{4}-\frac{52\!\cdots\!96}{38\!\cdots\!41}a^{2}+\frac{34\!\cdots\!83}{38\!\cdots\!41}$, $\frac{55\!\cdots\!77}{61\!\cdots\!56}a^{34}+\frac{37\!\cdots\!45}{61\!\cdots\!56}a^{32}+\frac{54\!\cdots\!23}{30\!\cdots\!28}a^{30}+\frac{14\!\cdots\!57}{49\!\cdots\!48}a^{28}+\frac{19\!\cdots\!07}{61\!\cdots\!56}a^{26}+\frac{29\!\cdots\!37}{12\!\cdots\!12}a^{24}+\frac{91\!\cdots\!49}{76\!\cdots\!82}a^{22}+\frac{51\!\cdots\!57}{12\!\cdots\!12}a^{20}+\frac{15\!\cdots\!73}{15\!\cdots\!64}a^{18}+\frac{55\!\cdots\!71}{30\!\cdots\!28}a^{16}+\frac{81\!\cdots\!52}{38\!\cdots\!41}a^{14}+\frac{24\!\cdots\!67}{15\!\cdots\!64}a^{12}+\frac{52\!\cdots\!21}{76\!\cdots\!82}a^{10}+\frac{33\!\cdots\!64}{38\!\cdots\!41}a^{8}-\frac{16\!\cdots\!77}{38\!\cdots\!41}a^{6}-\frac{12\!\cdots\!71}{76\!\cdots\!82}a^{4}-\frac{49\!\cdots\!38}{38\!\cdots\!41}a^{2}-\frac{49\!\cdots\!83}{38\!\cdots\!41}$, $\frac{18\!\cdots\!97}{98\!\cdots\!96}a^{34}+\frac{15\!\cdots\!73}{12\!\cdots\!12}a^{32}+\frac{92\!\cdots\!43}{24\!\cdots\!24}a^{30}+\frac{98\!\cdots\!83}{15\!\cdots\!64}a^{28}+\frac{17\!\cdots\!15}{24\!\cdots\!24}a^{26}+\frac{15\!\cdots\!17}{30\!\cdots\!28}a^{24}+\frac{16\!\cdots\!47}{61\!\cdots\!56}a^{22}+\frac{38\!\cdots\!06}{38\!\cdots\!41}a^{20}+\frac{16\!\cdots\!51}{61\!\cdots\!56}a^{18}+\frac{38\!\cdots\!63}{76\!\cdots\!82}a^{16}+\frac{10\!\cdots\!85}{15\!\cdots\!64}a^{14}+\frac{24\!\cdots\!12}{38\!\cdots\!41}a^{12}+\frac{32\!\cdots\!93}{76\!\cdots\!82}a^{10}+\frac{68\!\cdots\!96}{38\!\cdots\!41}a^{8}+\frac{18\!\cdots\!74}{38\!\cdots\!41}a^{6}+\frac{26\!\cdots\!08}{38\!\cdots\!41}a^{4}+\frac{16\!\cdots\!49}{38\!\cdots\!41}a^{2}+\frac{22\!\cdots\!77}{38\!\cdots\!41}$, $\frac{24\!\cdots\!97}{98\!\cdots\!96}a^{34}+\frac{20\!\cdots\!51}{12\!\cdots\!12}a^{32}+\frac{12\!\cdots\!87}{24\!\cdots\!24}a^{30}+\frac{25\!\cdots\!57}{30\!\cdots\!28}a^{28}+\frac{22\!\cdots\!67}{24\!\cdots\!24}a^{26}+\frac{19\!\cdots\!77}{30\!\cdots\!28}a^{24}+\frac{20\!\cdots\!15}{61\!\cdots\!56}a^{22}+\frac{14\!\cdots\!57}{12\!\cdots\!12}a^{20}+\frac{18\!\cdots\!21}{61\!\cdots\!56}a^{18}+\frac{16\!\cdots\!37}{30\!\cdots\!28}a^{16}+\frac{54\!\cdots\!23}{76\!\cdots\!82}a^{14}+\frac{19\!\cdots\!15}{30\!\cdots\!28}a^{12}+\frac{15\!\cdots\!00}{38\!\cdots\!41}a^{10}+\frac{12\!\cdots\!33}{76\!\cdots\!82}a^{8}+\frac{16\!\cdots\!68}{38\!\cdots\!41}a^{6}+\frac{45\!\cdots\!29}{76\!\cdots\!82}a^{4}+\frac{14\!\cdots\!63}{38\!\cdots\!41}a^{2}+\frac{24\!\cdots\!20}{38\!\cdots\!41}$, $\frac{18\!\cdots\!01}{98\!\cdots\!96}a^{34}+\frac{14\!\cdots\!55}{98\!\cdots\!96}a^{32}+\frac{12\!\cdots\!91}{24\!\cdots\!24}a^{30}+\frac{29\!\cdots\!11}{30\!\cdots\!28}a^{28}+\frac{29\!\cdots\!79}{24\!\cdots\!24}a^{26}+\frac{38\!\cdots\!45}{38\!\cdots\!41}a^{24}+\frac{36\!\cdots\!31}{61\!\cdots\!56}a^{22}+\frac{95\!\cdots\!69}{38\!\cdots\!41}a^{20}+\frac{45\!\cdots\!55}{61\!\cdots\!56}a^{18}+\frac{23\!\cdots\!81}{15\!\cdots\!64}a^{16}+\frac{35\!\cdots\!09}{15\!\cdots\!64}a^{14}+\frac{93\!\cdots\!68}{38\!\cdots\!41}a^{12}+\frac{13\!\cdots\!01}{76\!\cdots\!82}a^{10}+\frac{30\!\cdots\!52}{38\!\cdots\!41}a^{8}+\frac{87\!\cdots\!58}{38\!\cdots\!41}a^{6}+\frac{13\!\cdots\!96}{38\!\cdots\!41}a^{4}+\frac{98\!\cdots\!37}{38\!\cdots\!41}a^{2}+\frac{24\!\cdots\!24}{38\!\cdots\!41}$, $\frac{11\!\cdots\!89}{98\!\cdots\!96}a^{34}+\frac{98\!\cdots\!69}{12\!\cdots\!12}a^{32}+\frac{58\!\cdots\!19}{24\!\cdots\!24}a^{30}+\frac{31\!\cdots\!03}{76\!\cdots\!82}a^{28}+\frac{10\!\cdots\!71}{24\!\cdots\!24}a^{26}+\frac{80\!\cdots\!83}{24\!\cdots\!24}a^{24}+\frac{10\!\cdots\!91}{61\!\cdots\!56}a^{22}+\frac{18\!\cdots\!15}{30\!\cdots\!28}a^{20}+\frac{98\!\cdots\!59}{61\!\cdots\!56}a^{18}+\frac{90\!\cdots\!21}{30\!\cdots\!28}a^{16}+\frac{59\!\cdots\!79}{15\!\cdots\!64}a^{14}+\frac{13\!\cdots\!85}{38\!\cdots\!41}a^{12}+\frac{16\!\cdots\!83}{76\!\cdots\!82}a^{10}+\frac{12\!\cdots\!79}{15\!\cdots\!64}a^{8}+\frac{74\!\cdots\!50}{38\!\cdots\!41}a^{6}+\frac{10\!\cdots\!26}{38\!\cdots\!41}a^{4}+\frac{71\!\cdots\!69}{38\!\cdots\!41}a^{2}+\frac{17\!\cdots\!64}{38\!\cdots\!41}$, $\frac{16\!\cdots\!17}{98\!\cdots\!96}a^{34}+\frac{14\!\cdots\!17}{12\!\cdots\!12}a^{32}+\frac{84\!\cdots\!73}{24\!\cdots\!24}a^{30}+\frac{36\!\cdots\!17}{61\!\cdots\!56}a^{28}+\frac{15\!\cdots\!43}{24\!\cdots\!24}a^{26}+\frac{73\!\cdots\!41}{15\!\cdots\!64}a^{24}+\frac{15\!\cdots\!39}{61\!\cdots\!56}a^{22}+\frac{71\!\cdots\!63}{76\!\cdots\!82}a^{20}+\frac{15\!\cdots\!87}{61\!\cdots\!56}a^{18}+\frac{18\!\cdots\!34}{38\!\cdots\!41}a^{16}+\frac{10\!\cdots\!53}{15\!\cdots\!64}a^{14}+\frac{25\!\cdots\!50}{38\!\cdots\!41}a^{12}+\frac{34\!\cdots\!33}{76\!\cdots\!82}a^{10}+\frac{31\!\cdots\!79}{15\!\cdots\!64}a^{8}+\frac{21\!\cdots\!90}{38\!\cdots\!41}a^{6}+\frac{34\!\cdots\!10}{38\!\cdots\!41}a^{4}+\frac{21\!\cdots\!45}{38\!\cdots\!41}a^{2}+\frac{26\!\cdots\!76}{38\!\cdots\!41}$, $\frac{21\!\cdots\!85}{98\!\cdots\!96}a^{34}+\frac{32\!\cdots\!69}{24\!\cdots\!24}a^{32}+\frac{39\!\cdots\!07}{12\!\cdots\!12}a^{30}+\frac{48\!\cdots\!09}{12\!\cdots\!12}a^{28}+\frac{60\!\cdots\!51}{24\!\cdots\!24}a^{26}+\frac{10\!\cdots\!87}{61\!\cdots\!56}a^{24}-\frac{54\!\cdots\!65}{61\!\cdots\!56}a^{22}-\frac{11\!\cdots\!81}{15\!\cdots\!64}a^{20}-\frac{18\!\cdots\!89}{61\!\cdots\!56}a^{18}-\frac{12\!\cdots\!81}{15\!\cdots\!64}a^{16}-\frac{22\!\cdots\!83}{15\!\cdots\!64}a^{14}-\frac{66\!\cdots\!25}{38\!\cdots\!41}a^{12}-\frac{10\!\cdots\!03}{76\!\cdots\!82}a^{10}-\frac{27\!\cdots\!14}{38\!\cdots\!41}a^{8}-\frac{87\!\cdots\!74}{38\!\cdots\!41}a^{6}-\frac{29\!\cdots\!31}{76\!\cdots\!82}a^{4}-\frac{10\!\cdots\!95}{38\!\cdots\!41}a^{2}-\frac{26\!\cdots\!36}{38\!\cdots\!41}$, $\frac{19\!\cdots\!25}{98\!\cdots\!96}a^{34}+\frac{16\!\cdots\!33}{12\!\cdots\!12}a^{32}+\frac{10\!\cdots\!31}{24\!\cdots\!24}a^{30}+\frac{10\!\cdots\!53}{15\!\cdots\!64}a^{28}+\frac{19\!\cdots\!11}{24\!\cdots\!24}a^{26}+\frac{18\!\cdots\!61}{30\!\cdots\!28}a^{24}+\frac{19\!\cdots\!95}{61\!\cdots\!56}a^{22}+\frac{91\!\cdots\!25}{76\!\cdots\!82}a^{20}+\frac{19\!\cdots\!03}{61\!\cdots\!56}a^{18}+\frac{38\!\cdots\!51}{61\!\cdots\!56}a^{16}+\frac{13\!\cdots\!05}{15\!\cdots\!64}a^{14}+\frac{33\!\cdots\!97}{38\!\cdots\!41}a^{12}+\frac{44\!\cdots\!17}{76\!\cdots\!82}a^{10}+\frac{99\!\cdots\!07}{38\!\cdots\!41}a^{8}+\frac{26\!\cdots\!82}{38\!\cdots\!41}a^{6}+\frac{39\!\cdots\!92}{38\!\cdots\!41}a^{4}+\frac{24\!\cdots\!25}{38\!\cdots\!41}a^{2}+\frac{33\!\cdots\!12}{38\!\cdots\!41}$, $\frac{60\!\cdots\!21}{98\!\cdots\!96}a^{34}+\frac{20\!\cdots\!47}{49\!\cdots\!48}a^{32}+\frac{15\!\cdots\!59}{12\!\cdots\!12}a^{30}+\frac{50\!\cdots\!49}{24\!\cdots\!24}a^{28}+\frac{54\!\cdots\!73}{24\!\cdots\!24}a^{26}+\frac{39\!\cdots\!81}{24\!\cdots\!24}a^{24}+\frac{31\!\cdots\!21}{38\!\cdots\!41}a^{22}+\frac{36\!\cdots\!69}{12\!\cdots\!12}a^{20}+\frac{47\!\cdots\!99}{61\!\cdots\!56}a^{18}+\frac{44\!\cdots\!33}{30\!\cdots\!28}a^{16}+\frac{14\!\cdots\!03}{76\!\cdots\!82}a^{14}+\frac{14\!\cdots\!45}{76\!\cdots\!82}a^{12}+\frac{46\!\cdots\!47}{38\!\cdots\!41}a^{10}+\frac{40\!\cdots\!95}{76\!\cdots\!82}a^{8}+\frac{54\!\cdots\!21}{38\!\cdots\!41}a^{6}+\frac{16\!\cdots\!39}{76\!\cdots\!82}a^{4}+\frac{53\!\cdots\!00}{38\!\cdots\!41}a^{2}+\frac{64\!\cdots\!17}{38\!\cdots\!41}$, $\frac{56\!\cdots\!45}{98\!\cdots\!96}a^{34}+\frac{50\!\cdots\!87}{12\!\cdots\!12}a^{32}+\frac{32\!\cdots\!07}{24\!\cdots\!24}a^{30}+\frac{73\!\cdots\!83}{30\!\cdots\!28}a^{28}+\frac{70\!\cdots\!09}{24\!\cdots\!24}a^{26}+\frac{14\!\cdots\!23}{61\!\cdots\!56}a^{24}+\frac{40\!\cdots\!09}{30\!\cdots\!28}a^{22}+\frac{16\!\cdots\!11}{30\!\cdots\!28}a^{20}+\frac{97\!\cdots\!33}{61\!\cdots\!56}a^{18}+\frac{51\!\cdots\!51}{15\!\cdots\!64}a^{16}+\frac{77\!\cdots\!05}{15\!\cdots\!64}a^{14}+\frac{16\!\cdots\!77}{30\!\cdots\!28}a^{12}+\frac{30\!\cdots\!45}{76\!\cdots\!82}a^{10}+\frac{14\!\cdots\!49}{76\!\cdots\!82}a^{8}+\frac{22\!\cdots\!59}{38\!\cdots\!41}a^{6}+\frac{71\!\cdots\!41}{76\!\cdots\!82}a^{4}+\frac{23\!\cdots\!19}{38\!\cdots\!41}a^{2}+\frac{30\!\cdots\!28}{38\!\cdots\!41}$
| 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: | \( 28122649019657.055 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 28122649019657.055 \cdot 10893133}{2\cdot\sqrt{52733281945045886724167383478270850720626086921526306402773390818541568}}\cr\approx \mathstrut & 0.155372540556310 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512)
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 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512, 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 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512);
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 + 68*x^34 + 2018*x^32 + 34520*x^30 + 379376*x^28 + 2832128*x^26 + 14836664*x^24 + 55646624*x^22 + 151170256*x^20 + 298819648*x^18 + 428794592*x^16 + 442226304*x^14 + 321513984*x^12 + 159622144*x^10 + 51473664*x^8 + 9968640*x^6 + 1025280*x^4 + 46080*x^2 + 512);
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
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 $C_{36}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.3.361.1, 4.0.2048.2, 6.6.66724352.1, \(\Q(\zeta_{19})^+\), 12.0.145887695661298614272.69, 18.18.38713951190154487490850848768.1 |
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 | R | $36$ | $36$ | ${\href{/padicField/7.3.0.1}{3} }^{12}$ | ${\href{/padicField/11.12.0.1}{12} }^{3}$ | $36$ | ${\href{/padicField/17.9.0.1}{9} }^{4}$ | R | ${\href{/padicField/23.9.0.1}{9} }^{4}$ | $36$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{9}$ | $18^{2}$ | $36$ | $18^{2}$ | $36$ | $36$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $36$ | $4$ | $9$ | $99$ | |||
|
\(19\)
| Deg $36$ | $9$ | $4$ | $32$ |