Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649)
gp: K = bnfinit(y^36 - 29*y^33 + 584*y^30 - 5973*y^27 + 43132*y^24 - 106620*y^21 + 146403*y^18 - 601883*y^15 + 1334658*y^12 - 870682*y^9 + 395165*y^6 - 324478*y^3 + 117649, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649)
\( x^{36} - 29 x^{33} + 584 x^{30} - 5973 x^{27} + 43132 x^{24} - 106620 x^{21} + 146403 x^{18} + \cdots + 117649 \)
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: |
\(13401573218702027067604638251610483016128327492774416692842974769\)
\(\medspace = 3^{54}\cdot 19^{30}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.44\) | 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: | $3^{3/2}19^{5/6}\approx 60.437971963962546$ | ||
| Ramified primes: |
\(3\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(171=3^{2}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{171}(1,·)$, $\chi_{171}(134,·)$, $\chi_{171}(7,·)$, $\chi_{171}(8,·)$, $\chi_{171}(11,·)$, $\chi_{171}(140,·)$, $\chi_{171}(145,·)$, $\chi_{171}(20,·)$, $\chi_{171}(151,·)$, $\chi_{171}(26,·)$, $\chi_{171}(31,·)$, $\chi_{171}(160,·)$, $\chi_{171}(163,·)$, $\chi_{171}(164,·)$, $\chi_{171}(37,·)$, $\chi_{171}(170,·)$, $\chi_{171}(46,·)$, $\chi_{171}(49,·)$, $\chi_{171}(50,·)$, $\chi_{171}(56,·)$, $\chi_{171}(58,·)$, $\chi_{171}(64,·)$, $\chi_{171}(65,·)$, $\chi_{171}(68,·)$, $\chi_{171}(77,·)$, $\chi_{171}(83,·)$, $\chi_{171}(88,·)$, $\chi_{171}(94,·)$, $\chi_{171}(103,·)$, $\chi_{171}(106,·)$, $\chi_{171}(107,·)$, $\chi_{171}(113,·)$, $\chi_{171}(115,·)$, $\chi_{171}(121,·)$, $\chi_{171}(122,·)$, $\chi_{171}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{7}+\frac{1}{7}a$, $\frac{1}{7}a^{14}+\frac{1}{7}a^{8}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{15}+\frac{1}{7}a^{9}+\frac{1}{7}a^{3}$, $\frac{1}{7}a^{16}+\frac{1}{7}a^{10}+\frac{1}{7}a^{4}$, $\frac{1}{7}a^{17}+\frac{1}{7}a^{11}+\frac{1}{7}a^{5}$, $\frac{1}{7}a^{18}+\frac{1}{7}a^{12}+\frac{1}{7}a^{6}$, $\frac{1}{7}a^{19}-\frac{1}{7}a$, $\frac{1}{7}a^{20}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{21}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{22}-\frac{1}{7}a^{4}$, $\frac{1}{7}a^{23}-\frac{1}{7}a^{5}$, $\frac{1}{7}a^{24}-\frac{1}{7}a^{6}$, $\frac{1}{7}a^{25}-\frac{1}{7}a^{7}$, $\frac{1}{49}a^{26}+\frac{2}{49}a^{20}+\frac{3}{49}a^{14}+\frac{2}{49}a^{8}+\frac{1}{49}a^{2}$, $\frac{1}{111818}a^{27}-\frac{342}{7987}a^{24}+\frac{2143}{55909}a^{21}+\frac{15}{1141}a^{18}-\frac{2711}{55909}a^{15}+\frac{526}{1141}a^{12}+\frac{52103}{111818}a^{9}-\frac{834}{7987}a^{6}+\frac{5387}{55909}a^{3}+\frac{17}{326}$, $\frac{1}{111818}a^{28}-\frac{342}{7987}a^{25}+\frac{2143}{55909}a^{22}+\frac{15}{1141}a^{19}-\frac{2711}{55909}a^{16}+\frac{37}{1141}a^{13}+\frac{52103}{111818}a^{10}+\frac{3730}{7987}a^{7}+\frac{5387}{55909}a^{4}-\frac{859}{2282}a$, $\frac{1}{111818}a^{29}-\frac{16}{7987}a^{26}+\frac{2143}{55909}a^{23}-\frac{384}{7987}a^{20}-\frac{2711}{55909}a^{17}+\frac{96}{7987}a^{14}+\frac{52103}{111818}a^{11}+\frac{463}{1141}a^{8}+\frac{5387}{55909}a^{5}-\frac{5361}{15974}a^{2}$, $\frac{1}{1229998}a^{30}-\frac{1}{1229998}a^{27}-\frac{28538}{614999}a^{24}-\frac{19993}{614999}a^{21}+\frac{17428}{614999}a^{18}-\frac{12843}{614999}a^{15}+\frac{77975}{1229998}a^{12}+\frac{210985}{1229998}a^{9}-\frac{98437}{614999}a^{6}+\frac{176637}{1229998}a^{3}-\frac{773}{3586}$, $\frac{1}{1229998}a^{31}-\frac{1}{1229998}a^{28}-\frac{28538}{614999}a^{25}-\frac{19993}{614999}a^{22}+\frac{17428}{614999}a^{19}-\frac{12843}{614999}a^{16}+\frac{77975}{1229998}a^{13}+\frac{210985}{1229998}a^{10}-\frac{98437}{614999}a^{7}+\frac{176637}{1229998}a^{4}-\frac{773}{3586}a$, $\frac{1}{1229998}a^{32}-\frac{1}{1229998}a^{29}-\frac{3436}{614999}a^{26}-\frac{19993}{614999}a^{23}-\frac{20225}{614999}a^{20}-\frac{12843}{614999}a^{17}+\frac{52873}{1229998}a^{14}+\frac{210985}{1229998}a^{11}-\frac{136090}{614999}a^{8}+\frac{176637}{1229998}a^{5}-\frac{30705}{175714}a^{2}$, $\frac{1}{10\!\cdots\!18}a^{33}-\frac{68\!\cdots\!65}{50\!\cdots\!59}a^{30}-\frac{25\!\cdots\!13}{10\!\cdots\!91}a^{27}-\frac{35\!\cdots\!80}{50\!\cdots\!59}a^{24}+\frac{17\!\cdots\!39}{50\!\cdots\!59}a^{21}-\frac{27\!\cdots\!29}{50\!\cdots\!59}a^{18}-\frac{71\!\cdots\!45}{10\!\cdots\!18}a^{15}+\frac{14\!\cdots\!64}{50\!\cdots\!59}a^{12}+\frac{80\!\cdots\!27}{50\!\cdots\!59}a^{9}+\frac{44\!\cdots\!79}{10\!\cdots\!18}a^{6}-\frac{67\!\cdots\!65}{50\!\cdots\!59}a^{3}+\frac{66\!\cdots\!51}{14\!\cdots\!13}$, $\frac{1}{70\!\cdots\!26}a^{34}-\frac{86\!\cdots\!61}{63\!\cdots\!66}a^{31}+\frac{47\!\cdots\!09}{35\!\cdots\!13}a^{28}+\frac{37\!\cdots\!21}{35\!\cdots\!13}a^{25}-\frac{24\!\cdots\!03}{35\!\cdots\!13}a^{22}+\frac{17\!\cdots\!19}{35\!\cdots\!13}a^{19}+\frac{47\!\cdots\!55}{70\!\cdots\!26}a^{16}+\frac{11\!\cdots\!05}{70\!\cdots\!26}a^{13}-\frac{48\!\cdots\!09}{35\!\cdots\!13}a^{10}-\frac{22\!\cdots\!69}{70\!\cdots\!26}a^{7}-\frac{30\!\cdots\!53}{70\!\cdots\!26}a^{4}-\frac{39\!\cdots\!08}{10\!\cdots\!91}a$, $\frac{1}{49\!\cdots\!82}a^{35}-\frac{86\!\cdots\!61}{44\!\cdots\!62}a^{32}-\frac{57\!\cdots\!48}{24\!\cdots\!91}a^{29}-\frac{16\!\cdots\!22}{24\!\cdots\!91}a^{26}-\frac{16\!\cdots\!46}{24\!\cdots\!91}a^{23}+\frac{15\!\cdots\!06}{24\!\cdots\!91}a^{20}+\frac{21\!\cdots\!81}{49\!\cdots\!82}a^{17}+\frac{26\!\cdots\!95}{49\!\cdots\!82}a^{14}-\frac{10\!\cdots\!47}{24\!\cdots\!91}a^{11}+\frac{23\!\cdots\!27}{49\!\cdots\!82}a^{8}+\frac{23\!\cdots\!15}{49\!\cdots\!82}a^{5}-\frac{16\!\cdots\!81}{71\!\cdots\!37}a^{2}$
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
$C_{3}\times C_{6}\times C_{42}$, which has order $756$ (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: |
\( \frac{923190657524195357}{64955138739974823821467} a^{35} - \frac{376216007488709476203}{909371942359647533500538} a^{32} + \frac{540067001267869045925}{64955138739974823821467} a^{29} - \frac{38645480084335274688870}{454685971179823766750269} a^{26} + \frac{39407404381382012210629}{64955138739974823821467} a^{23} - \frac{656462217340336671701788}{454685971179823766750269} a^{20} + \frac{86292983585252471704543}{64955138739974823821467} a^{17} - \frac{7454330131825724013869939}{909371942359647533500538} a^{14} + \frac{1159233791555800453201708}{64955138739974823821467} a^{11} - \frac{1021065831999859772015020}{454685971179823766750269} a^{8} + \frac{5889749282309807780379}{2651230152652033625366} a^{5} - \frac{205446867111044266614083}{64955138739974823821467} a^{2} \)
(order $18$)
| 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: |
$\frac{15\!\cdots\!87}{44\!\cdots\!62}a^{35}-\frac{10\!\cdots\!69}{64\!\cdots\!67}a^{34}-\frac{52\!\cdots\!43}{44\!\cdots\!62}a^{32}+\frac{20\!\cdots\!78}{45\!\cdots\!69}a^{31}+\frac{55\!\cdots\!93}{22\!\cdots\!81}a^{29}-\frac{16\!\cdots\!73}{18\!\cdots\!62}a^{28}-\frac{66\!\cdots\!17}{22\!\cdots\!81}a^{26}+\frac{39\!\cdots\!92}{45\!\cdots\!69}a^{25}+\frac{51\!\cdots\!99}{22\!\cdots\!81}a^{23}-\frac{39\!\cdots\!64}{64\!\cdots\!67}a^{22}-\frac{20\!\cdots\!00}{22\!\cdots\!81}a^{20}+\frac{49\!\cdots\!30}{45\!\cdots\!69}a^{19}+\frac{32\!\cdots\!47}{44\!\cdots\!62}a^{17}-\frac{88\!\cdots\!67}{64\!\cdots\!67}a^{16}-\frac{13\!\cdots\!39}{44\!\cdots\!62}a^{14}+\frac{37\!\cdots\!00}{45\!\cdots\!69}a^{13}+\frac{26\!\cdots\!94}{22\!\cdots\!81}a^{11}-\frac{16\!\cdots\!33}{12\!\cdots\!34}a^{10}-\frac{18\!\cdots\!73}{44\!\cdots\!62}a^{8}+\frac{10\!\cdots\!35}{45\!\cdots\!69}a^{7}+\frac{12\!\cdots\!73}{44\!\cdots\!62}a^{5}-\frac{29\!\cdots\!68}{13\!\cdots\!83}a^{4}-\frac{25\!\cdots\!09}{13\!\cdots\!83}a^{2}+\frac{56\!\cdots\!23}{18\!\cdots\!62}a$, $\frac{92\!\cdots\!57}{64\!\cdots\!67}a^{35}-\frac{37\!\cdots\!03}{90\!\cdots\!38}a^{32}+\frac{54\!\cdots\!25}{64\!\cdots\!67}a^{29}-\frac{38\!\cdots\!70}{45\!\cdots\!69}a^{26}+\frac{39\!\cdots\!29}{64\!\cdots\!67}a^{23}-\frac{65\!\cdots\!88}{45\!\cdots\!69}a^{20}+\frac{86\!\cdots\!43}{64\!\cdots\!67}a^{17}-\frac{74\!\cdots\!39}{90\!\cdots\!38}a^{14}+\frac{11\!\cdots\!08}{64\!\cdots\!67}a^{11}-\frac{10\!\cdots\!20}{45\!\cdots\!69}a^{8}+\frac{58\!\cdots\!79}{26\!\cdots\!66}a^{5}-\frac{20\!\cdots\!83}{64\!\cdots\!67}a^{2}-1$, $\frac{21\!\cdots\!09}{64\!\cdots\!67}a^{33}-\frac{45\!\cdots\!38}{50\!\cdots\!59}a^{30}+\frac{18\!\cdots\!89}{10\!\cdots\!91}a^{27}-\frac{84\!\cdots\!92}{50\!\cdots\!59}a^{24}+\frac{82\!\cdots\!24}{71\!\cdots\!37}a^{21}-\frac{86\!\cdots\!84}{50\!\cdots\!59}a^{18}+\frac{18\!\cdots\!37}{71\!\cdots\!37}a^{15}-\frac{78\!\cdots\!80}{50\!\cdots\!59}a^{12}+\frac{84\!\cdots\!36}{71\!\cdots\!37}a^{9}-\frac{21\!\cdots\!85}{50\!\cdots\!59}a^{6}+\frac{61\!\cdots\!48}{14\!\cdots\!13}a^{3}+\frac{18\!\cdots\!27}{14\!\cdots\!13}$, $\frac{36\!\cdots\!05}{14\!\cdots\!74}a^{33}-\frac{76\!\cdots\!03}{10\!\cdots\!18}a^{30}+\frac{15\!\cdots\!69}{10\!\cdots\!18}a^{27}-\frac{81\!\cdots\!02}{50\!\cdots\!59}a^{24}+\frac{59\!\cdots\!96}{50\!\cdots\!59}a^{21}-\frac{16\!\cdots\!02}{50\!\cdots\!59}a^{18}+\frac{33\!\cdots\!67}{10\!\cdots\!18}a^{15}-\frac{16\!\cdots\!77}{10\!\cdots\!18}a^{12}+\frac{41\!\cdots\!85}{10\!\cdots\!18}a^{9}-\frac{14\!\cdots\!69}{10\!\cdots\!18}a^{6}+\frac{12\!\cdots\!93}{10\!\cdots\!18}a^{3}-\frac{17\!\cdots\!05}{26\!\cdots\!66}$, $\frac{12\!\cdots\!85}{49\!\cdots\!82}a^{35}-\frac{17\!\cdots\!58}{24\!\cdots\!91}a^{32}+\frac{34\!\cdots\!72}{24\!\cdots\!91}a^{29}-\frac{31\!\cdots\!16}{24\!\cdots\!91}a^{26}+\frac{21\!\cdots\!31}{24\!\cdots\!91}a^{23}-\frac{29\!\cdots\!40}{24\!\cdots\!91}a^{20}+\frac{11\!\cdots\!57}{49\!\cdots\!82}a^{17}-\frac{29\!\cdots\!39}{24\!\cdots\!91}a^{14}+\frac{33\!\cdots\!41}{24\!\cdots\!91}a^{11}-\frac{35\!\cdots\!51}{49\!\cdots\!82}a^{8}+\frac{62\!\cdots\!99}{24\!\cdots\!91}a^{5}-\frac{11\!\cdots\!57}{92\!\cdots\!81}a^{2}$, $\frac{10\!\cdots\!98}{50\!\cdots\!59}a^{33}-\frac{59\!\cdots\!09}{10\!\cdots\!18}a^{30}+\frac{11\!\cdots\!21}{10\!\cdots\!18}a^{27}-\frac{59\!\cdots\!15}{50\!\cdots\!59}a^{24}+\frac{38\!\cdots\!68}{45\!\cdots\!69}a^{21}-\frac{89\!\cdots\!97}{50\!\cdots\!59}a^{18}+\frac{15\!\cdots\!40}{71\!\cdots\!37}a^{15}-\frac{11\!\cdots\!27}{10\!\cdots\!18}a^{12}+\frac{20\!\cdots\!51}{90\!\cdots\!38}a^{9}-\frac{38\!\cdots\!32}{50\!\cdots\!59}a^{6}+\frac{57\!\cdots\!73}{10\!\cdots\!18}a^{3}-\frac{17\!\cdots\!33}{29\!\cdots\!26}$, $\frac{19\!\cdots\!45}{70\!\cdots\!26}a^{34}-\frac{53\!\cdots\!07}{70\!\cdots\!26}a^{31}+\frac{10\!\cdots\!13}{70\!\cdots\!26}a^{28}-\frac{49\!\cdots\!67}{35\!\cdots\!13}a^{25}+\frac{33\!\cdots\!76}{35\!\cdots\!13}a^{22}-\frac{43\!\cdots\!49}{35\!\cdots\!13}a^{19}+\frac{14\!\cdots\!29}{70\!\cdots\!26}a^{16}-\frac{87\!\cdots\!27}{70\!\cdots\!26}a^{13}+\frac{94\!\cdots\!55}{70\!\cdots\!26}a^{10}-\frac{17\!\cdots\!43}{70\!\cdots\!26}a^{7}-\frac{98\!\cdots\!99}{70\!\cdots\!26}a^{4}+\frac{78\!\cdots\!95}{20\!\cdots\!82}a$, $\frac{21\!\cdots\!07}{12\!\cdots\!34}a^{35}-\frac{14\!\cdots\!35}{12\!\cdots\!34}a^{34}+\frac{26\!\cdots\!85}{90\!\cdots\!38}a^{33}-\frac{23\!\cdots\!37}{50\!\cdots\!59}a^{32}+\frac{13\!\cdots\!95}{50\!\cdots\!59}a^{31}-\frac{72\!\cdots\!31}{90\!\cdots\!38}a^{30}+\frac{68\!\cdots\!11}{71\!\cdots\!37}a^{29}-\frac{10\!\cdots\!69}{20\!\cdots\!82}a^{28}+\frac{71\!\cdots\!23}{45\!\cdots\!69}a^{27}-\frac{49\!\cdots\!58}{50\!\cdots\!59}a^{26}+\frac{17\!\cdots\!18}{50\!\cdots\!59}a^{25}-\frac{68\!\cdots\!83}{45\!\cdots\!69}a^{24}+\frac{50\!\cdots\!76}{71\!\cdots\!37}a^{23}-\frac{14\!\cdots\!24}{71\!\cdots\!37}a^{22}+\frac{46\!\cdots\!42}{45\!\cdots\!69}a^{21}-\frac{86\!\cdots\!31}{50\!\cdots\!59}a^{20}-\frac{32\!\cdots\!01}{50\!\cdots\!59}a^{19}-\frac{73\!\cdots\!88}{45\!\cdots\!69}a^{18}+\frac{22\!\cdots\!51}{14\!\cdots\!74}a^{17}-\frac{77\!\cdots\!63}{14\!\cdots\!74}a^{16}+\frac{22\!\cdots\!93}{90\!\cdots\!38}a^{15}-\frac{47\!\cdots\!20}{50\!\cdots\!59}a^{14}+\frac{13\!\cdots\!34}{50\!\cdots\!59}a^{13}-\frac{12\!\cdots\!73}{90\!\cdots\!38}a^{12}+\frac{15\!\cdots\!76}{71\!\cdots\!37}a^{11}+\frac{14\!\cdots\!15}{14\!\cdots\!74}a^{10}+\frac{85\!\cdots\!10}{45\!\cdots\!69}a^{9}-\frac{26\!\cdots\!55}{10\!\cdots\!18}a^{8}+\frac{76\!\cdots\!35}{10\!\cdots\!18}a^{7}-\frac{60\!\cdots\!91}{90\!\cdots\!38}a^{6}+\frac{37\!\cdots\!52}{14\!\cdots\!13}a^{5}-\frac{10\!\cdots\!86}{14\!\cdots\!13}a^{4}+\frac{54\!\cdots\!95}{90\!\cdots\!38}a^{3}-\frac{31\!\cdots\!05}{71\!\cdots\!37}a^{2}-\frac{52\!\cdots\!21}{20\!\cdots\!82}a-\frac{53\!\cdots\!78}{13\!\cdots\!83}$, $\frac{48\!\cdots\!15}{44\!\cdots\!62}a^{35}+\frac{65\!\cdots\!15}{35\!\cdots\!13}a^{34}-\frac{12\!\cdots\!38}{64\!\cdots\!67}a^{33}-\frac{65\!\cdots\!02}{22\!\cdots\!81}a^{32}-\frac{19\!\cdots\!35}{35\!\cdots\!13}a^{31}+\frac{57\!\cdots\!27}{10\!\cdots\!18}a^{30}+\frac{12\!\cdots\!82}{22\!\cdots\!81}a^{29}+\frac{48\!\cdots\!65}{42\!\cdots\!02}a^{28}-\frac{11\!\cdots\!05}{10\!\cdots\!91}a^{27}-\frac{12\!\cdots\!13}{22\!\cdots\!81}a^{26}-\frac{41\!\cdots\!32}{35\!\cdots\!13}a^{25}+\frac{59\!\cdots\!02}{50\!\cdots\!59}a^{24}+\frac{83\!\cdots\!48}{22\!\cdots\!81}a^{23}+\frac{30\!\cdots\!50}{35\!\cdots\!13}a^{22}-\frac{60\!\cdots\!77}{71\!\cdots\!37}a^{21}-\frac{11\!\cdots\!12}{22\!\cdots\!81}a^{20}-\frac{82\!\cdots\!95}{35\!\cdots\!13}a^{19}+\frac{10\!\cdots\!27}{50\!\cdots\!59}a^{18}+\frac{26\!\cdots\!51}{44\!\cdots\!62}a^{17}+\frac{89\!\cdots\!75}{35\!\cdots\!13}a^{16}-\frac{13\!\cdots\!78}{71\!\cdots\!37}a^{15}-\frac{11\!\cdots\!86}{22\!\cdots\!81}a^{14}-\frac{42\!\cdots\!65}{35\!\cdots\!13}a^{13}+\frac{11\!\cdots\!43}{10\!\cdots\!18}a^{12}+\frac{13\!\cdots\!50}{22\!\cdots\!81}a^{11}+\frac{20\!\cdots\!35}{70\!\cdots\!26}a^{10}-\frac{18\!\cdots\!58}{71\!\cdots\!37}a^{9}+\frac{80\!\cdots\!13}{44\!\cdots\!62}a^{8}-\frac{44\!\cdots\!85}{35\!\cdots\!13}a^{7}+\frac{15\!\cdots\!75}{50\!\cdots\!59}a^{6}+\frac{43\!\cdots\!90}{22\!\cdots\!81}a^{5}+\frac{27\!\cdots\!00}{31\!\cdots\!83}a^{4}-\frac{90\!\cdots\!19}{29\!\cdots\!26}a^{3}-\frac{78\!\cdots\!42}{64\!\cdots\!67}a^{2}-\frac{15\!\cdots\!35}{20\!\cdots\!82}a+\frac{81\!\cdots\!21}{14\!\cdots\!13}$, $\frac{15\!\cdots\!87}{44\!\cdots\!62}a^{35}-\frac{18\!\cdots\!19}{35\!\cdots\!13}a^{34}-\frac{12\!\cdots\!67}{50\!\cdots\!59}a^{33}-\frac{52\!\cdots\!43}{44\!\cdots\!62}a^{32}+\frac{50\!\cdots\!07}{35\!\cdots\!13}a^{31}+\frac{66\!\cdots\!35}{10\!\cdots\!18}a^{30}+\frac{55\!\cdots\!93}{22\!\cdots\!81}a^{29}-\frac{10\!\cdots\!12}{35\!\cdots\!13}a^{28}-\frac{64\!\cdots\!56}{50\!\cdots\!59}a^{27}-\frac{66\!\cdots\!17}{22\!\cdots\!81}a^{26}+\frac{94\!\cdots\!34}{35\!\cdots\!13}a^{25}+\frac{56\!\cdots\!74}{50\!\cdots\!59}a^{24}+\frac{51\!\cdots\!99}{22\!\cdots\!81}a^{23}-\frac{65\!\cdots\!08}{35\!\cdots\!13}a^{22}-\frac{52\!\cdots\!70}{71\!\cdots\!37}a^{21}-\frac{20\!\cdots\!00}{22\!\cdots\!81}a^{20}+\frac{95\!\cdots\!32}{35\!\cdots\!13}a^{19}+\frac{90\!\cdots\!80}{50\!\cdots\!59}a^{18}+\frac{32\!\cdots\!47}{44\!\cdots\!62}a^{17}-\frac{15\!\cdots\!80}{35\!\cdots\!13}a^{16}-\frac{75\!\cdots\!67}{50\!\cdots\!59}a^{15}-\frac{13\!\cdots\!39}{44\!\cdots\!62}a^{14}+\frac{88\!\cdots\!41}{35\!\cdots\!13}a^{13}+\frac{96\!\cdots\!05}{10\!\cdots\!18}a^{12}+\frac{26\!\cdots\!94}{22\!\cdots\!81}a^{11}-\frac{11\!\cdots\!70}{35\!\cdots\!13}a^{10}-\frac{22\!\cdots\!58}{50\!\cdots\!59}a^{9}-\frac{18\!\cdots\!73}{44\!\cdots\!62}a^{8}+\frac{32\!\cdots\!53}{35\!\cdots\!13}a^{7}+\frac{14\!\cdots\!73}{50\!\cdots\!59}a^{6}+\frac{12\!\cdots\!73}{44\!\cdots\!62}a^{5}-\frac{32\!\cdots\!65}{31\!\cdots\!83}a^{4}+\frac{24\!\cdots\!07}{10\!\cdots\!18}a^{3}-\frac{25\!\cdots\!09}{13\!\cdots\!83}a^{2}+\frac{67\!\cdots\!02}{10\!\cdots\!91}a-\frac{81\!\cdots\!08}{13\!\cdots\!83}$, $\frac{23\!\cdots\!01}{12\!\cdots\!34}a^{35}-\frac{69\!\cdots\!03}{63\!\cdots\!66}a^{34}-\frac{29\!\cdots\!77}{14\!\cdots\!74}a^{33}-\frac{25\!\cdots\!81}{50\!\cdots\!59}a^{32}+\frac{18\!\cdots\!07}{63\!\cdots\!66}a^{31}+\frac{40\!\cdots\!18}{71\!\cdots\!37}a^{30}+\frac{74\!\cdots\!22}{71\!\cdots\!37}a^{29}-\frac{35\!\cdots\!29}{63\!\cdots\!66}a^{28}-\frac{11\!\cdots\!87}{10\!\cdots\!18}a^{27}-\frac{53\!\cdots\!06}{50\!\cdots\!59}a^{26}+\frac{15\!\cdots\!58}{31\!\cdots\!83}a^{25}+\frac{76\!\cdots\!56}{71\!\cdots\!37}a^{24}+\frac{54\!\cdots\!44}{71\!\cdots\!37}a^{23}-\frac{10\!\cdots\!46}{31\!\cdots\!83}a^{22}-\frac{37\!\cdots\!86}{50\!\cdots\!59}a^{21}-\frac{90\!\cdots\!48}{50\!\cdots\!59}a^{20}+\frac{72\!\cdots\!10}{31\!\cdots\!83}a^{19}+\frac{88\!\cdots\!09}{71\!\cdots\!37}a^{18}+\frac{23\!\cdots\!25}{14\!\cdots\!74}a^{17}-\frac{46\!\cdots\!03}{63\!\cdots\!66}a^{16}-\frac{22\!\cdots\!11}{10\!\cdots\!18}a^{15}-\frac{51\!\cdots\!46}{50\!\cdots\!59}a^{14}+\frac{27\!\cdots\!05}{63\!\cdots\!66}a^{13}+\frac{72\!\cdots\!90}{71\!\cdots\!37}a^{12}+\frac{15\!\cdots\!06}{71\!\cdots\!37}a^{11}-\frac{12\!\cdots\!97}{63\!\cdots\!66}a^{10}-\frac{14\!\cdots\!47}{10\!\cdots\!18}a^{9}-\frac{28\!\cdots\!05}{10\!\cdots\!18}a^{8}+\frac{50\!\cdots\!57}{63\!\cdots\!66}a^{7}+\frac{15\!\cdots\!73}{14\!\cdots\!74}a^{6}+\frac{40\!\cdots\!90}{14\!\cdots\!13}a^{5}+\frac{60\!\cdots\!89}{63\!\cdots\!66}a^{4}-\frac{20\!\cdots\!65}{45\!\cdots\!69}a^{3}-\frac{74\!\cdots\!40}{71\!\cdots\!37}a^{2}+\frac{80\!\cdots\!69}{26\!\cdots\!66}a+\frac{93\!\cdots\!75}{29\!\cdots\!26}$, $\frac{11\!\cdots\!15}{49\!\cdots\!82}a^{35}-\frac{35\!\cdots\!59}{63\!\cdots\!66}a^{34}-\frac{28\!\cdots\!54}{64\!\cdots\!67}a^{33}-\frac{31\!\cdots\!65}{49\!\cdots\!82}a^{32}+\frac{10\!\cdots\!85}{63\!\cdots\!66}a^{31}+\frac{59\!\cdots\!53}{50\!\cdots\!59}a^{30}+\frac{62\!\cdots\!55}{49\!\cdots\!82}a^{29}-\frac{11\!\cdots\!05}{31\!\cdots\!83}a^{28}-\frac{47\!\cdots\!87}{20\!\cdots\!82}a^{27}-\frac{29\!\cdots\!20}{24\!\cdots\!91}a^{26}+\frac{12\!\cdots\!86}{31\!\cdots\!83}a^{25}+\frac{65\!\cdots\!44}{30\!\cdots\!93}a^{24}+\frac{20\!\cdots\!40}{24\!\cdots\!91}a^{23}-\frac{89\!\cdots\!90}{31\!\cdots\!83}a^{22}-\frac{10\!\cdots\!54}{71\!\cdots\!37}a^{21}-\frac{31\!\cdots\!30}{24\!\cdots\!91}a^{20}+\frac{27\!\cdots\!00}{31\!\cdots\!83}a^{19}+\frac{83\!\cdots\!89}{50\!\cdots\!59}a^{18}+\frac{99\!\cdots\!95}{49\!\cdots\!82}a^{17}-\frac{40\!\cdots\!63}{63\!\cdots\!66}a^{16}-\frac{23\!\cdots\!82}{71\!\cdots\!37}a^{15}-\frac{54\!\cdots\!45}{49\!\cdots\!82}a^{14}+\frac{25\!\cdots\!95}{63\!\cdots\!66}a^{13}+\frac{97\!\cdots\!65}{50\!\cdots\!59}a^{12}+\frac{72\!\cdots\!35}{49\!\cdots\!82}a^{11}-\frac{34\!\cdots\!10}{31\!\cdots\!83}a^{10}-\frac{26\!\cdots\!51}{14\!\cdots\!74}a^{9}-\frac{25\!\cdots\!45}{49\!\cdots\!82}a^{8}+\frac{93\!\cdots\!33}{63\!\cdots\!66}a^{7}+\frac{26\!\cdots\!60}{50\!\cdots\!59}a^{6}+\frac{45\!\cdots\!13}{49\!\cdots\!82}a^{5}-\frac{20\!\cdots\!25}{63\!\cdots\!66}a^{4}-\frac{77\!\cdots\!53}{14\!\cdots\!13}a^{3}-\frac{62\!\cdots\!35}{26\!\cdots\!66}a^{2}+\frac{25\!\cdots\!70}{92\!\cdots\!81}a+\frac{12\!\cdots\!89}{29\!\cdots\!26}$, $\frac{81\!\cdots\!87}{49\!\cdots\!82}a^{35}-\frac{39\!\cdots\!11}{12\!\cdots\!34}a^{34}-\frac{21\!\cdots\!25}{49\!\cdots\!82}a^{32}+\frac{83\!\cdots\!17}{10\!\cdots\!18}a^{31}+\frac{20\!\cdots\!45}{24\!\cdots\!91}a^{29}-\frac{47\!\cdots\!97}{29\!\cdots\!26}a^{28}-\frac{18\!\cdots\!93}{24\!\cdots\!91}a^{26}+\frac{76\!\cdots\!20}{50\!\cdots\!59}a^{25}+\frac{12\!\cdots\!06}{24\!\cdots\!91}a^{23}-\frac{75\!\cdots\!01}{71\!\cdots\!37}a^{22}-\frac{74\!\cdots\!16}{24\!\cdots\!91}a^{20}+\frac{70\!\cdots\!26}{50\!\cdots\!59}a^{19}+\frac{21\!\cdots\!89}{30\!\cdots\!14}a^{17}-\frac{34\!\cdots\!19}{14\!\cdots\!74}a^{16}-\frac{32\!\cdots\!87}{49\!\cdots\!82}a^{14}+\frac{14\!\cdots\!11}{10\!\cdots\!18}a^{13}+\frac{58\!\cdots\!18}{24\!\cdots\!91}a^{11}-\frac{22\!\cdots\!01}{14\!\cdots\!74}a^{10}+\frac{19\!\cdots\!55}{49\!\cdots\!82}a^{8}+\frac{38\!\cdots\!85}{10\!\cdots\!18}a^{7}+\frac{50\!\cdots\!95}{44\!\cdots\!62}a^{5}-\frac{11\!\cdots\!91}{29\!\cdots\!26}a^{4}-\frac{30\!\cdots\!02}{71\!\cdots\!37}a^{2}+\frac{82\!\cdots\!55}{20\!\cdots\!82}a-1$, $\frac{11\!\cdots\!17}{44\!\cdots\!62}a^{35}+\frac{18\!\cdots\!19}{35\!\cdots\!13}a^{34}-\frac{12\!\cdots\!67}{50\!\cdots\!59}a^{33}-\frac{31\!\cdots\!51}{44\!\cdots\!62}a^{32}-\frac{50\!\cdots\!07}{35\!\cdots\!13}a^{31}+\frac{66\!\cdots\!35}{10\!\cdots\!18}a^{30}+\frac{31\!\cdots\!57}{22\!\cdots\!81}a^{29}+\frac{10\!\cdots\!12}{35\!\cdots\!13}a^{28}-\frac{64\!\cdots\!56}{50\!\cdots\!59}a^{27}-\frac{31\!\cdots\!43}{22\!\cdots\!81}a^{26}-\frac{94\!\cdots\!34}{35\!\cdots\!13}a^{25}+\frac{56\!\cdots\!74}{50\!\cdots\!59}a^{24}+\frac{21\!\cdots\!95}{22\!\cdots\!81}a^{23}+\frac{65\!\cdots\!08}{35\!\cdots\!13}a^{22}-\frac{52\!\cdots\!70}{71\!\cdots\!37}a^{21}-\frac{43\!\cdots\!24}{22\!\cdots\!81}a^{20}-\frac{95\!\cdots\!32}{35\!\cdots\!13}a^{19}+\frac{90\!\cdots\!80}{50\!\cdots\!59}a^{18}+\frac{86\!\cdots\!49}{44\!\cdots\!62}a^{17}+\frac{15\!\cdots\!80}{35\!\cdots\!13}a^{16}-\frac{75\!\cdots\!67}{50\!\cdots\!59}a^{15}-\frac{36\!\cdots\!41}{27\!\cdots\!74}a^{14}-\frac{88\!\cdots\!41}{35\!\cdots\!13}a^{13}+\frac{96\!\cdots\!05}{10\!\cdots\!18}a^{12}+\frac{52\!\cdots\!94}{22\!\cdots\!81}a^{11}+\frac{11\!\cdots\!70}{35\!\cdots\!13}a^{10}-\frac{22\!\cdots\!58}{50\!\cdots\!59}a^{9}-\frac{19\!\cdots\!47}{44\!\cdots\!62}a^{8}-\frac{32\!\cdots\!53}{35\!\cdots\!13}a^{7}+\frac{14\!\cdots\!73}{50\!\cdots\!59}a^{6}+\frac{18\!\cdots\!33}{44\!\cdots\!62}a^{5}+\frac{32\!\cdots\!65}{31\!\cdots\!83}a^{4}+\frac{24\!\cdots\!07}{10\!\cdots\!18}a^{3}-\frac{28\!\cdots\!25}{64\!\cdots\!67}a^{2}-\frac{67\!\cdots\!02}{10\!\cdots\!91}a-\frac{81\!\cdots\!08}{13\!\cdots\!83}$, $\frac{11\!\cdots\!17}{44\!\cdots\!62}a^{35}-\frac{39\!\cdots\!11}{12\!\cdots\!34}a^{34}+\frac{28\!\cdots\!54}{64\!\cdots\!67}a^{33}-\frac{31\!\cdots\!51}{44\!\cdots\!62}a^{32}+\frac{83\!\cdots\!17}{10\!\cdots\!18}a^{31}-\frac{59\!\cdots\!53}{50\!\cdots\!59}a^{30}+\frac{31\!\cdots\!57}{22\!\cdots\!81}a^{29}-\frac{47\!\cdots\!97}{29\!\cdots\!26}a^{28}+\frac{47\!\cdots\!87}{20\!\cdots\!82}a^{27}-\frac{31\!\cdots\!43}{22\!\cdots\!81}a^{26}+\frac{76\!\cdots\!20}{50\!\cdots\!59}a^{25}-\frac{65\!\cdots\!44}{30\!\cdots\!93}a^{24}+\frac{21\!\cdots\!95}{22\!\cdots\!81}a^{23}-\frac{75\!\cdots\!01}{71\!\cdots\!37}a^{22}+\frac{10\!\cdots\!54}{71\!\cdots\!37}a^{21}-\frac{43\!\cdots\!24}{22\!\cdots\!81}a^{20}+\frac{70\!\cdots\!26}{50\!\cdots\!59}a^{19}-\frac{83\!\cdots\!89}{50\!\cdots\!59}a^{18}+\frac{86\!\cdots\!49}{44\!\cdots\!62}a^{17}-\frac{34\!\cdots\!19}{14\!\cdots\!74}a^{16}+\frac{23\!\cdots\!82}{71\!\cdots\!37}a^{15}-\frac{36\!\cdots\!41}{27\!\cdots\!74}a^{14}+\frac{14\!\cdots\!11}{10\!\cdots\!18}a^{13}-\frac{97\!\cdots\!65}{50\!\cdots\!59}a^{12}+\frac{52\!\cdots\!94}{22\!\cdots\!81}a^{11}-\frac{22\!\cdots\!01}{14\!\cdots\!74}a^{10}+\frac{26\!\cdots\!51}{14\!\cdots\!74}a^{9}-\frac{19\!\cdots\!47}{44\!\cdots\!62}a^{8}+\frac{38\!\cdots\!85}{10\!\cdots\!18}a^{7}-\frac{26\!\cdots\!60}{50\!\cdots\!59}a^{6}+\frac{18\!\cdots\!33}{44\!\cdots\!62}a^{5}-\frac{11\!\cdots\!91}{29\!\cdots\!26}a^{4}+\frac{77\!\cdots\!53}{14\!\cdots\!13}a^{3}-\frac{28\!\cdots\!25}{64\!\cdots\!67}a^{2}+\frac{82\!\cdots\!55}{20\!\cdots\!82}a-\frac{12\!\cdots\!89}{29\!\cdots\!26}$, $\frac{81\!\cdots\!87}{49\!\cdots\!82}a^{35}+\frac{15\!\cdots\!09}{70\!\cdots\!26}a^{34}+\frac{26\!\cdots\!85}{90\!\cdots\!38}a^{33}-\frac{21\!\cdots\!25}{49\!\cdots\!82}a^{32}-\frac{43\!\cdots\!95}{70\!\cdots\!26}a^{31}-\frac{72\!\cdots\!31}{90\!\cdots\!38}a^{30}+\frac{20\!\cdots\!45}{24\!\cdots\!91}a^{29}+\frac{86\!\cdots\!27}{70\!\cdots\!26}a^{28}+\frac{71\!\cdots\!23}{45\!\cdots\!69}a^{27}-\frac{18\!\cdots\!93}{24\!\cdots\!91}a^{26}-\frac{41\!\cdots\!94}{35\!\cdots\!13}a^{25}-\frac{68\!\cdots\!83}{45\!\cdots\!69}a^{24}+\frac{12\!\cdots\!06}{24\!\cdots\!91}a^{23}+\frac{28\!\cdots\!59}{35\!\cdots\!13}a^{22}+\frac{46\!\cdots\!42}{45\!\cdots\!69}a^{21}-\frac{74\!\cdots\!16}{24\!\cdots\!91}a^{20}-\frac{46\!\cdots\!50}{35\!\cdots\!13}a^{19}-\frac{73\!\cdots\!88}{45\!\cdots\!69}a^{18}+\frac{21\!\cdots\!89}{30\!\cdots\!14}a^{17}+\frac{14\!\cdots\!29}{70\!\cdots\!26}a^{16}+\frac{22\!\cdots\!93}{90\!\cdots\!38}a^{15}-\frac{32\!\cdots\!87}{49\!\cdots\!82}a^{14}-\frac{78\!\cdots\!05}{70\!\cdots\!26}a^{13}-\frac{12\!\cdots\!73}{90\!\cdots\!38}a^{12}+\frac{58\!\cdots\!18}{24\!\cdots\!91}a^{11}+\frac{10\!\cdots\!91}{70\!\cdots\!26}a^{10}+\frac{85\!\cdots\!10}{45\!\cdots\!69}a^{9}+\frac{19\!\cdots\!55}{49\!\cdots\!82}a^{8}-\frac{38\!\cdots\!11}{70\!\cdots\!26}a^{7}-\frac{60\!\cdots\!91}{90\!\cdots\!38}a^{6}+\frac{50\!\cdots\!95}{44\!\cdots\!62}a^{5}+\frac{44\!\cdots\!39}{70\!\cdots\!26}a^{4}+\frac{54\!\cdots\!95}{90\!\cdots\!38}a^{3}-\frac{30\!\cdots\!02}{71\!\cdots\!37}a^{2}-\frac{66\!\cdots\!37}{26\!\cdots\!66}a-\frac{26\!\cdots\!12}{13\!\cdots\!83}$, $\frac{78\!\cdots\!89}{44\!\cdots\!62}a^{35}-\frac{25\!\cdots\!07}{79\!\cdots\!18}a^{34}+\frac{87\!\cdots\!90}{45\!\cdots\!69}a^{33}-\frac{11\!\cdots\!45}{22\!\cdots\!81}a^{32}+\frac{11\!\cdots\!27}{10\!\cdots\!18}a^{31}-\frac{22\!\cdots\!61}{45\!\cdots\!69}a^{30}+\frac{24\!\cdots\!68}{22\!\cdots\!81}a^{29}-\frac{47\!\cdots\!95}{20\!\cdots\!82}a^{28}+\frac{88\!\cdots\!67}{90\!\cdots\!38}a^{27}-\frac{25\!\cdots\!47}{22\!\cdots\!81}a^{26}+\frac{13\!\cdots\!60}{50\!\cdots\!59}a^{25}-\frac{38\!\cdots\!93}{45\!\cdots\!69}a^{24}+\frac{18\!\cdots\!46}{22\!\cdots\!81}a^{23}-\frac{15\!\cdots\!51}{71\!\cdots\!37}a^{22}+\frac{25\!\cdots\!70}{45\!\cdots\!69}a^{21}-\frac{52\!\cdots\!12}{22\!\cdots\!81}a^{20}+\frac{40\!\cdots\!67}{50\!\cdots\!59}a^{19}-\frac{10\!\cdots\!07}{45\!\cdots\!69}a^{18}+\frac{91\!\cdots\!45}{44\!\cdots\!62}a^{17}-\frac{62\!\cdots\!29}{14\!\cdots\!74}a^{16}+\frac{48\!\cdots\!41}{45\!\cdots\!69}a^{15}-\frac{25\!\cdots\!25}{22\!\cdots\!81}a^{14}+\frac{28\!\cdots\!81}{10\!\cdots\!18}a^{13}-\frac{33\!\cdots\!79}{45\!\cdots\!69}a^{12}+\frac{66\!\cdots\!38}{22\!\cdots\!81}a^{11}-\frac{15\!\cdots\!13}{14\!\cdots\!74}a^{10}+\frac{99\!\cdots\!11}{90\!\cdots\!38}a^{9}-\frac{28\!\cdots\!33}{44\!\cdots\!62}a^{8}+\frac{78\!\cdots\!35}{10\!\cdots\!18}a^{7}+\frac{48\!\cdots\!09}{45\!\cdots\!69}a^{6}+\frac{55\!\cdots\!63}{22\!\cdots\!81}a^{5}-\frac{22\!\cdots\!81}{29\!\cdots\!26}a^{4}+\frac{55\!\cdots\!90}{64\!\cdots\!67}a^{3}-\frac{20\!\cdots\!48}{39\!\cdots\!09}a^{2}+\frac{39\!\cdots\!03}{20\!\cdots\!82}a+\frac{10\!\cdots\!09}{26\!\cdots\!66}$
| 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: | \( 2007721725946302.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 2007721725946302.5 \cdot 756}{18\cdot\sqrt{13401573218702027067604638251610483016128327492774416692842974769}}\cr\approx \mathstrut & 0.169672925387799 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649)
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 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649, 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 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649);
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 - 29*x^33 + 584*x^30 - 5973*x^27 + 43132*x^24 - 106620*x^21 + 146403*x^18 - 601883*x^15 + 1334658*x^12 - 870682*x^9 + 395165*x^6 - 324478*x^3 + 117649);
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)
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_6^2$ |
| Character table for $C_6^2$ |
Intermediate fields
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 | ${\href{/padicField/2.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{12}$ | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{18}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $36$ | $6$ | $6$ | $54$ | |||
|
\(19\)
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |