Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201)
gp: K = bnfinit(y^36 - 394*y^30 + 118583*y^24 - 14205984*y^18 + 1297088703*y^12 - 4312188797*y^6 + 13841287201, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201)
\( x^{36} - 394x^{30} + 118583x^{24} - 14205984x^{18} + 1297088703x^{12} - 4312188797x^{6} + 13841287201 \)
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: |
\(19575552194003861659818831697024491945650963873549328070183080689664\)
\(\medspace = 2^{36}\cdot 3^{54}\cdot 19^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(74.00\) | 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: | $2\cdot 3^{3/2}19^{2/3}\approx 73.99702820503963$ | ||
| Ramified primes: |
\(2\), \(3\), \(19\)
| 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: | \(684=2^{2}\cdot 3^{2}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{684}(1,·)$, $\chi_{684}(235,·)$, $\chi_{684}(391,·)$, $\chi_{684}(11,·)$, $\chi_{684}(653,·)$, $\chi_{684}(121,·)$, $\chi_{684}(277,·)$, $\chi_{684}(539,·)$, $\chi_{684}(197,·)$, $\chi_{684}(163,·)$, $\chi_{684}(305,·)$, $\chi_{684}(425,·)$, $\chi_{684}(7,·)$, $\chi_{684}(49,·)$, $\chi_{684}(115,·)$, $\chi_{684}(311,·)$, $\chi_{684}(571,·)$, $\chi_{684}(191,·)$, $\chi_{684}(577,·)$, $\chi_{684}(83,·)$, $\chi_{684}(581,·)$, $\chi_{684}(457,·)$, $\chi_{684}(77,·)$, $\chi_{684}(463,·)$, $\chi_{684}(419,·)$, $\chi_{684}(343,·)$, $\chi_{684}(349,·)$, $\chi_{684}(353,·)$, $\chi_{684}(229,·)$, $\chi_{684}(619,·)$, $\chi_{684}(647,·)$, $\chi_{684}(239,·)$, $\chi_{684}(467,·)$, $\chi_{684}(505,·)$, $\chi_{684}(125,·)$, $\chi_{684}(533,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{1}{7}a^{7}+\frac{1}{7}a$, $\frac{1}{7}a^{14}-\frac{1}{7}a^{8}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{15}-\frac{1}{7}a^{9}+\frac{1}{7}a^{3}$, $\frac{1}{7}a^{16}-\frac{1}{7}a^{10}+\frac{1}{7}a^{4}$, $\frac{1}{7}a^{17}-\frac{1}{7}a^{11}+\frac{1}{7}a^{5}$, $\frac{1}{7}a^{18}-\frac{1}{7}a^{12}+\frac{1}{7}a^{6}$, $\frac{1}{7}a^{19}+\frac{1}{7}a$, $\frac{1}{7}a^{20}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{21}+\frac{1}{7}a^{3}$, $\frac{1}{7}a^{22}+\frac{1}{7}a^{4}$, $\frac{1}{7}a^{23}+\frac{1}{7}a^{5}$, $\frac{1}{189847}a^{24}-\frac{6796}{189847}a^{18}+\frac{64476}{189847}a^{12}+\frac{28030}{189847}a^{6}-\frac{3873}{27121}$, $\frac{1}{189847}a^{25}-\frac{6796}{189847}a^{19}+\frac{1462}{27121}a^{13}+\frac{82272}{189847}a^{7}-\frac{81353}{189847}a$, $\frac{1}{1328929}a^{26}-\frac{33917}{1328929}a^{20}+\frac{37355}{1328929}a^{14}-\frac{324543}{1328929}a^{8}+\frac{108494}{1328929}a^{2}$, $\frac{1}{9302503}a^{27}-\frac{413611}{9302503}a^{21}-\frac{532186}{9302503}a^{15}-\frac{2412860}{9302503}a^{9}+\frac{1817117}{9302503}a^{3}$, $\frac{1}{65117521}a^{28}-\frac{4400398}{65117521}a^{22}+\frac{3454601}{65117521}a^{16}+\frac{21507862}{65117521}a^{10}-\frac{7485386}{65117521}a^{4}$, $\frac{1}{455822647}a^{29}-\frac{13702901}{455822647}a^{23}-\frac{5847902}{455822647}a^{17}-\frac{34307156}{455822647}a^{11}-\frac{91207913}{455822647}a^{5}$, $\frac{1}{48\!\cdots\!99}a^{30}+\frac{24\!\cdots\!26}{48\!\cdots\!99}a^{24}-\frac{14\!\cdots\!11}{48\!\cdots\!99}a^{18}+\frac{23\!\cdots\!05}{48\!\cdots\!99}a^{12}+\frac{36\!\cdots\!10}{48\!\cdots\!99}a^{6}-\frac{12\!\cdots\!66}{41\!\cdots\!51}$, $\frac{1}{33\!\cdots\!93}a^{31}+\frac{53\!\cdots\!60}{33\!\cdots\!93}a^{25}-\frac{18\!\cdots\!89}{33\!\cdots\!93}a^{19}+\frac{59\!\cdots\!92}{91\!\cdots\!89}a^{13}-\frac{96\!\cdots\!82}{33\!\cdots\!93}a^{7}-\frac{10\!\cdots\!94}{28\!\cdots\!57}a$, $\frac{1}{23\!\cdots\!51}a^{32}+\frac{53\!\cdots\!60}{23\!\cdots\!51}a^{26}-\frac{67\!\cdots\!88}{23\!\cdots\!51}a^{20}+\frac{32\!\cdots\!46}{64\!\cdots\!23}a^{14}+\frac{14\!\cdots\!13}{23\!\cdots\!51}a^{8}-\frac{93\!\cdots\!49}{28\!\cdots\!57}a^{2}$, $\frac{1}{16\!\cdots\!57}a^{33}+\frac{53\!\cdots\!60}{16\!\cdots\!57}a^{27}+\frac{27\!\cdots\!05}{16\!\cdots\!57}a^{21}+\frac{30\!\cdots\!13}{44\!\cdots\!61}a^{15}+\frac{62\!\cdots\!87}{16\!\cdots\!57}a^{9}-\frac{10\!\cdots\!88}{28\!\cdots\!57}a^{3}$, $\frac{1}{11\!\cdots\!99}a^{34}+\frac{53\!\cdots\!60}{11\!\cdots\!99}a^{28}+\frac{74\!\cdots\!58}{11\!\cdots\!99}a^{22}+\frac{95\!\cdots\!36}{31\!\cdots\!27}a^{16}+\frac{37\!\cdots\!50}{11\!\cdots\!99}a^{10}+\frac{64\!\cdots\!30}{20\!\cdots\!99}a^{4}$, $\frac{1}{81\!\cdots\!93}a^{35}+\frac{53\!\cdots\!60}{81\!\cdots\!93}a^{29}+\frac{57\!\cdots\!29}{81\!\cdots\!93}a^{23}+\frac{14\!\cdots\!19}{22\!\cdots\!89}a^{17}-\frac{12\!\cdots\!21}{81\!\cdots\!93}a^{11}-\frac{16\!\cdots\!26}{14\!\cdots\!93}a^{5}$
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
$C_{3}\times C_{12}\times C_{444}$, which has order $15984$ (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: |
\( \frac{745606568130638905}{8157411590985903361344769393} a^{35} - \frac{291275732977210271517}{8157411590985903361344769393} a^{29} + \frac{87473322363672887039805}{8157411590985903361344769393} a^{23} - \frac{10308275970137477235408725}{8157411590985903361344769393} a^{17} + \frac{934315988642476886763757963}{8157411590985903361344769393} a^{11} - \frac{56104081927926830}{4125474770350538551} a^{5} \)
(order $36$)
| 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: |
$\frac{36153969142}{99\!\cdots\!51}a^{34}-\frac{681395583810464}{33\!\cdots\!93}a^{31}-\frac{10881335336969}{99\!\cdots\!51}a^{28}+\frac{26\!\cdots\!40}{33\!\cdots\!93}a^{25}+\frac{29\!\cdots\!67}{99\!\cdots\!51}a^{22}-\frac{79\!\cdots\!84}{33\!\cdots\!93}a^{19}-\frac{11\!\cdots\!29}{99\!\cdots\!51}a^{16}+\frac{94\!\cdots\!80}{33\!\cdots\!93}a^{13}+\frac{164803096743971}{41\!\cdots\!51}a^{10}-\frac{86\!\cdots\!65}{33\!\cdots\!93}a^{7}+\frac{44\!\cdots\!11}{99\!\cdots\!51}a^{4}+\frac{12\!\cdots\!04}{41\!\cdots\!51}a$, $\frac{13\!\cdots\!56}{11\!\cdots\!99}a^{34}-\frac{39\!\cdots\!10}{23\!\cdots\!51}a^{32}-\frac{50\!\cdots\!35}{11\!\cdots\!99}a^{28}+\frac{15\!\cdots\!61}{23\!\cdots\!51}a^{26}+\frac{15\!\cdots\!36}{11\!\cdots\!99}a^{22}-\frac{46\!\cdots\!10}{23\!\cdots\!51}a^{20}-\frac{17\!\cdots\!20}{11\!\cdots\!99}a^{16}+\frac{54\!\cdots\!50}{23\!\cdots\!51}a^{14}+\frac{16\!\cdots\!52}{11\!\cdots\!99}a^{10}-\frac{49\!\cdots\!71}{23\!\cdots\!51}a^{8}-\frac{68\!\cdots\!12}{41\!\cdots\!51}a^{4}+\frac{10\!\cdots\!80}{41\!\cdots\!51}a^{2}$, $\frac{36153969142}{99\!\cdots\!51}a^{34}-\frac{38\!\cdots\!44}{23\!\cdots\!51}a^{32}-\frac{10881335336969}{99\!\cdots\!51}a^{28}+\frac{15\!\cdots\!24}{23\!\cdots\!51}a^{26}+\frac{29\!\cdots\!67}{99\!\cdots\!51}a^{22}-\frac{45\!\cdots\!68}{23\!\cdots\!51}a^{20}-\frac{11\!\cdots\!29}{99\!\cdots\!51}a^{16}+\frac{54\!\cdots\!33}{23\!\cdots\!51}a^{14}+\frac{164803096743971}{41\!\cdots\!51}a^{10}-\frac{49\!\cdots\!88}{23\!\cdots\!51}a^{8}+\frac{44\!\cdots\!11}{99\!\cdots\!51}a^{4}+\frac{14\!\cdots\!88}{20\!\cdots\!99}a^{2}$, $\frac{72\!\cdots\!59}{81\!\cdots\!93}a^{35}-\frac{28\!\cdots\!20}{81\!\cdots\!93}a^{29}+\frac{85\!\cdots\!40}{81\!\cdots\!93}a^{23}-\frac{10\!\cdots\!48}{81\!\cdots\!93}a^{17}+\frac{93\!\cdots\!40}{81\!\cdots\!93}a^{11}-\frac{26\!\cdots\!40}{69\!\cdots\!57}a^{5}+1$, $\frac{74\!\cdots\!05}{81\!\cdots\!93}a^{35}-\frac{29\!\cdots\!17}{81\!\cdots\!93}a^{29}+\frac{87\!\cdots\!05}{81\!\cdots\!93}a^{23}-\frac{10\!\cdots\!25}{81\!\cdots\!93}a^{17}+\frac{93\!\cdots\!63}{81\!\cdots\!93}a^{11}-\frac{56\!\cdots\!30}{41\!\cdots\!51}a^{5}+1$, $\frac{48\!\cdots\!47}{16\!\cdots\!57}a^{33}-\frac{19\!\cdots\!72}{16\!\cdots\!57}a^{27}+\frac{57\!\cdots\!54}{16\!\cdots\!57}a^{21}-\frac{68\!\cdots\!78}{16\!\cdots\!57}a^{15}+\frac{63\!\cdots\!14}{16\!\cdots\!57}a^{9}-\frac{17\!\cdots\!14}{14\!\cdots\!93}a^{3}$, $\frac{143299072257608}{48\!\cdots\!99}a^{30}-\frac{59\!\cdots\!86}{48\!\cdots\!99}a^{24}+\frac{17\!\cdots\!27}{48\!\cdots\!99}a^{18}-\frac{22\!\cdots\!68}{48\!\cdots\!99}a^{12}+\frac{19\!\cdots\!07}{48\!\cdots\!99}a^{6}-\frac{55\!\cdots\!57}{41\!\cdots\!51}$, $\frac{30\!\cdots\!86}{23\!\cdots\!51}a^{32}+\frac{30466886810400}{48\!\cdots\!99}a^{30}-\frac{12\!\cdots\!15}{23\!\cdots\!51}a^{26}-\frac{10\!\cdots\!48}{48\!\cdots\!99}a^{24}+\frac{36\!\cdots\!13}{23\!\cdots\!51}a^{20}+\frac{30\!\cdots\!86}{48\!\cdots\!99}a^{18}-\frac{42\!\cdots\!33}{23\!\cdots\!51}a^{14}-\frac{32\!\cdots\!81}{48\!\cdots\!99}a^{12}+\frac{38\!\cdots\!94}{23\!\cdots\!51}a^{8}+\frac{33\!\cdots\!26}{48\!\cdots\!99}a^{6}+\frac{26\!\cdots\!07}{20\!\cdots\!99}a^{2}+\frac{31\!\cdots\!25}{41\!\cdots\!51}$, $\frac{17\!\cdots\!59}{23\!\cdots\!51}a^{32}-\frac{132948264266337}{48\!\cdots\!99}a^{30}-\frac{68\!\cdots\!63}{23\!\cdots\!51}a^{26}+\frac{50\!\cdots\!63}{48\!\cdots\!99}a^{24}+\frac{20\!\cdots\!99}{23\!\cdots\!51}a^{20}-\frac{15\!\cdots\!32}{48\!\cdots\!99}a^{18}-\frac{24\!\cdots\!20}{23\!\cdots\!51}a^{14}+\frac{16\!\cdots\!76}{48\!\cdots\!99}a^{12}+\frac{22\!\cdots\!20}{23\!\cdots\!51}a^{8}-\frac{15\!\cdots\!38}{48\!\cdots\!99}a^{6}-\frac{49\!\cdots\!19}{20\!\cdots\!99}a^{2}-\frac{14\!\cdots\!95}{41\!\cdots\!51}$, $\frac{15\!\cdots\!56}{81\!\cdots\!93}a^{35}+\frac{33\!\cdots\!92}{11\!\cdots\!99}a^{34}+\frac{24\!\cdots\!81}{16\!\cdots\!57}a^{33}+\frac{13\!\cdots\!73}{23\!\cdots\!51}a^{32}+\frac{25\!\cdots\!22}{33\!\cdots\!93}a^{31}+\frac{415170036960906}{48\!\cdots\!99}a^{30}-\frac{60\!\cdots\!81}{81\!\cdots\!93}a^{29}-\frac{12\!\cdots\!32}{11\!\cdots\!99}a^{28}-\frac{95\!\cdots\!86}{16\!\cdots\!57}a^{27}-\frac{51\!\cdots\!42}{23\!\cdots\!51}a^{26}-\frac{10\!\cdots\!11}{33\!\cdots\!93}a^{25}-\frac{16\!\cdots\!28}{48\!\cdots\!99}a^{24}+\frac{18\!\cdots\!44}{81\!\cdots\!93}a^{23}+\frac{38\!\cdots\!37}{11\!\cdots\!99}a^{22}+\frac{28\!\cdots\!36}{16\!\cdots\!57}a^{21}+\frac{15\!\cdots\!05}{23\!\cdots\!51}a^{20}+\frac{30\!\cdots\!31}{33\!\cdots\!93}a^{19}+\frac{49\!\cdots\!00}{48\!\cdots\!99}a^{18}-\frac{20\!\cdots\!77}{81\!\cdots\!93}a^{17}-\frac{45\!\cdots\!51}{11\!\cdots\!99}a^{16}-\frac{32\!\cdots\!43}{16\!\cdots\!57}a^{15}-\frac{18\!\cdots\!79}{23\!\cdots\!51}a^{14}-\frac{35\!\cdots\!20}{33\!\cdots\!93}a^{13}-\frac{58\!\cdots\!89}{48\!\cdots\!99}a^{12}+\frac{18\!\cdots\!99}{81\!\cdots\!93}a^{11}+\frac{40\!\cdots\!08}{11\!\cdots\!99}a^{10}+\frac{28\!\cdots\!12}{16\!\cdots\!57}a^{9}+\frac{16\!\cdots\!46}{23\!\cdots\!51}a^{8}+\frac{32\!\cdots\!11}{33\!\cdots\!93}a^{7}+\frac{53\!\cdots\!21}{48\!\cdots\!99}a^{6}+\frac{81\!\cdots\!11}{69\!\cdots\!57}a^{5}+\frac{40\!\cdots\!43}{99\!\cdots\!51}a^{4}+\frac{19\!\cdots\!05}{14\!\cdots\!93}a^{3}+\frac{25\!\cdots\!54}{20\!\cdots\!99}a^{2}-\frac{85\!\cdots\!04}{28\!\cdots\!57}a+\frac{38\!\cdots\!14}{41\!\cdots\!51}$, $\frac{19\!\cdots\!81}{81\!\cdots\!93}a^{35}+\frac{16\!\cdots\!83}{11\!\cdots\!99}a^{34}+\frac{94\!\cdots\!47}{16\!\cdots\!57}a^{33}-\frac{38\!\cdots\!24}{23\!\cdots\!51}a^{32}-\frac{681395583810464}{33\!\cdots\!93}a^{31}+\frac{61992700971150}{48\!\cdots\!99}a^{30}-\frac{74\!\cdots\!11}{81\!\cdots\!93}a^{29}-\frac{63\!\cdots\!11}{11\!\cdots\!99}a^{28}-\frac{37\!\cdots\!68}{16\!\cdots\!57}a^{27}+\frac{15\!\cdots\!34}{23\!\cdots\!51}a^{26}+\frac{26\!\cdots\!40}{33\!\cdots\!93}a^{25}-\frac{22\!\cdots\!30}{48\!\cdots\!99}a^{24}+\frac{22\!\cdots\!35}{81\!\cdots\!93}a^{23}+\frac{19\!\cdots\!68}{11\!\cdots\!99}a^{22}+\frac{11\!\cdots\!69}{16\!\cdots\!57}a^{21}-\frac{45\!\cdots\!91}{23\!\cdots\!51}a^{20}-\frac{79\!\cdots\!84}{33\!\cdots\!93}a^{19}+\frac{67\!\cdots\!85}{48\!\cdots\!99}a^{18}-\frac{23\!\cdots\!18}{81\!\cdots\!93}a^{17}-\frac{23\!\cdots\!71}{11\!\cdots\!99}a^{16}-\frac{14\!\cdots\!31}{16\!\cdots\!57}a^{15}+\frac{54\!\cdots\!43}{23\!\cdots\!51}a^{14}+\frac{94\!\cdots\!80}{33\!\cdots\!93}a^{13}-\frac{69\!\cdots\!61}{48\!\cdots\!99}a^{12}+\frac{20\!\cdots\!98}{81\!\cdots\!93}a^{11}+\frac{20\!\cdots\!03}{11\!\cdots\!99}a^{10}+\frac{13\!\cdots\!19}{16\!\cdots\!57}a^{9}-\frac{49\!\cdots\!78}{23\!\cdots\!51}a^{8}-\frac{86\!\cdots\!65}{33\!\cdots\!93}a^{7}+\frac{74\!\cdots\!85}{48\!\cdots\!99}a^{6}+\frac{26\!\cdots\!50}{69\!\cdots\!57}a^{5}-\frac{49\!\cdots\!81}{99\!\cdots\!51}a^{4}-\frac{15\!\cdots\!54}{14\!\cdots\!93}a^{3}-\frac{25\!\cdots\!31}{20\!\cdots\!99}a^{2}+\frac{42\!\cdots\!55}{41\!\cdots\!51}a+\frac{61\!\cdots\!67}{41\!\cdots\!51}$, $\frac{64\!\cdots\!30}{81\!\cdots\!93}a^{35}+\frac{11\!\cdots\!92}{11\!\cdots\!99}a^{34}+\frac{49059800046}{14\!\cdots\!93}a^{33}-\frac{360276747229859}{64\!\cdots\!23}a^{32}-\frac{30\!\cdots\!31}{33\!\cdots\!93}a^{31}-\frac{828143736290466}{48\!\cdots\!99}a^{30}-\frac{25\!\cdots\!32}{81\!\cdots\!93}a^{29}-\frac{45\!\cdots\!42}{11\!\cdots\!99}a^{28}-\frac{14765630123997}{14\!\cdots\!93}a^{27}+\frac{52\!\cdots\!65}{23\!\cdots\!51}a^{26}+\frac{11\!\cdots\!80}{33\!\cdots\!93}a^{25}+\frac{32\!\cdots\!61}{48\!\cdots\!99}a^{24}+\frac{76\!\cdots\!24}{81\!\cdots\!93}a^{23}+\frac{13\!\cdots\!52}{11\!\cdots\!99}a^{22}+\frac{40\!\cdots\!41}{14\!\cdots\!93}a^{21}-\frac{15\!\cdots\!47}{23\!\cdots\!51}a^{20}-\frac{35\!\cdots\!64}{33\!\cdots\!93}a^{19}-\frac{97\!\cdots\!46}{48\!\cdots\!99}a^{18}-\frac{92\!\cdots\!82}{81\!\cdots\!93}a^{17}-\frac{16\!\cdots\!40}{11\!\cdots\!99}a^{16}-\frac{16\!\cdots\!77}{14\!\cdots\!93}a^{15}+\frac{19\!\cdots\!53}{23\!\cdots\!51}a^{14}+\frac{42\!\cdots\!78}{33\!\cdots\!93}a^{13}+\frac{11\!\cdots\!70}{48\!\cdots\!99}a^{12}+\frac{84\!\cdots\!84}{81\!\cdots\!93}a^{11}+\frac{14\!\cdots\!66}{11\!\cdots\!99}a^{10}+\frac{15\!\cdots\!61}{41\!\cdots\!51}a^{9}-\frac{17\!\cdots\!66}{23\!\cdots\!51}a^{8}-\frac{39\!\cdots\!45}{33\!\cdots\!93}a^{7}-\frac{10\!\cdots\!43}{48\!\cdots\!99}a^{6}-\frac{23\!\cdots\!84}{69\!\cdots\!57}a^{5}-\frac{61\!\cdots\!84}{41\!\cdots\!51}a^{4}+\frac{52\!\cdots\!66}{14\!\cdots\!93}a^{3}+\frac{62\!\cdots\!67}{20\!\cdots\!99}a^{2}+\frac{86\!\cdots\!08}{28\!\cdots\!57}a+\frac{10\!\cdots\!32}{41\!\cdots\!51}$, $\frac{47\!\cdots\!09}{11\!\cdots\!99}a^{34}-\frac{15\!\cdots\!63}{33\!\cdots\!93}a^{31}-\frac{17\!\cdots\!25}{11\!\cdots\!99}a^{28}+\frac{61\!\cdots\!61}{33\!\cdots\!93}a^{25}+\frac{52\!\cdots\!47}{11\!\cdots\!99}a^{22}-\frac{18\!\cdots\!52}{33\!\cdots\!93}a^{19}-\frac{15\!\cdots\!80}{31\!\cdots\!27}a^{16}+\frac{21\!\cdots\!92}{33\!\cdots\!93}a^{13}+\frac{46\!\cdots\!42}{11\!\cdots\!99}a^{10}-\frac{19\!\cdots\!51}{33\!\cdots\!93}a^{7}+\frac{98\!\cdots\!69}{99\!\cdots\!51}a^{4}-\frac{48\!\cdots\!69}{28\!\cdots\!57}a$, $\frac{15\!\cdots\!62}{81\!\cdots\!93}a^{35}+\frac{13\!\cdots\!22}{11\!\cdots\!99}a^{34}+\frac{30\!\cdots\!18}{16\!\cdots\!57}a^{33}+\frac{89\!\cdots\!90}{23\!\cdots\!51}a^{32}+\frac{25\!\cdots\!79}{33\!\cdots\!93}a^{31}+\frac{567892625611532}{48\!\cdots\!99}a^{30}-\frac{59\!\cdots\!98}{81\!\cdots\!93}a^{29}-\frac{53\!\cdots\!22}{11\!\cdots\!99}a^{28}-\frac{11\!\cdots\!57}{16\!\cdots\!57}a^{27}-\frac{34\!\cdots\!15}{23\!\cdots\!51}a^{26}-\frac{97\!\cdots\!24}{33\!\cdots\!93}a^{25}-\frac{22\!\cdots\!65}{48\!\cdots\!99}a^{24}+\frac{17\!\cdots\!56}{81\!\cdots\!93}a^{23}+\frac{15\!\cdots\!85}{11\!\cdots\!99}a^{22}+\frac{35\!\cdots\!77}{16\!\cdots\!57}a^{21}+\frac{10\!\cdots\!82}{23\!\cdots\!51}a^{20}+\frac{29\!\cdots\!97}{33\!\cdots\!93}a^{19}+\frac{67\!\cdots\!71}{48\!\cdots\!99}a^{18}-\frac{20\!\cdots\!74}{81\!\cdots\!93}a^{17}-\frac{18\!\cdots\!87}{11\!\cdots\!99}a^{16}-\frac{41\!\cdots\!02}{16\!\cdots\!57}a^{15}-\frac{12\!\cdots\!44}{23\!\cdots\!51}a^{14}-\frac{33\!\cdots\!42}{33\!\cdots\!93}a^{13}-\frac{81\!\cdots\!48}{48\!\cdots\!99}a^{12}+\frac{50\!\cdots\!46}{22\!\cdots\!89}a^{11}+\frac{16\!\cdots\!85}{11\!\cdots\!99}a^{10}+\frac{36\!\cdots\!66}{16\!\cdots\!57}a^{9}+\frac{11\!\cdots\!92}{23\!\cdots\!51}a^{8}+\frac{30\!\cdots\!36}{33\!\cdots\!93}a^{7}+\frac{72\!\cdots\!37}{48\!\cdots\!99}a^{6}+\frac{54\!\cdots\!87}{69\!\cdots\!57}a^{5}+\frac{55\!\cdots\!04}{99\!\cdots\!51}a^{4}+\frac{73\!\cdots\!44}{14\!\cdots\!93}a^{3}+\frac{25\!\cdots\!20}{20\!\cdots\!99}a^{2}+\frac{27\!\cdots\!34}{28\!\cdots\!57}a-\frac{60\!\cdots\!05}{41\!\cdots\!51}$, $\frac{75\!\cdots\!74}{81\!\cdots\!93}a^{35}-\frac{20\!\cdots\!37}{11\!\cdots\!99}a^{34}+\frac{94\!\cdots\!47}{16\!\cdots\!57}a^{33}-\frac{789181251755958}{23\!\cdots\!51}a^{32}-\frac{149405144180329}{33\!\cdots\!93}a^{31}+\frac{61992700971150}{48\!\cdots\!99}a^{30}-\frac{30\!\cdots\!70}{81\!\cdots\!93}a^{29}+\frac{87\!\cdots\!74}{11\!\cdots\!99}a^{28}-\frac{37\!\cdots\!68}{16\!\cdots\!57}a^{27}+\frac{82\!\cdots\!57}{64\!\cdots\!23}a^{26}+\frac{58\!\cdots\!64}{33\!\cdots\!93}a^{25}-\frac{22\!\cdots\!30}{48\!\cdots\!99}a^{24}+\frac{90\!\cdots\!64}{81\!\cdots\!93}a^{23}-\frac{27\!\cdots\!94}{11\!\cdots\!99}a^{22}+\frac{11\!\cdots\!69}{16\!\cdots\!57}a^{21}-\frac{92\!\cdots\!55}{23\!\cdots\!51}a^{20}-\frac{17\!\cdots\!98}{33\!\cdots\!93}a^{19}+\frac{67\!\cdots\!85}{48\!\cdots\!99}a^{18}-\frac{29\!\cdots\!51}{22\!\cdots\!89}a^{17}+\frac{36\!\cdots\!68}{11\!\cdots\!99}a^{16}-\frac{14\!\cdots\!31}{16\!\cdots\!57}a^{15}+\frac{11\!\cdots\!00}{23\!\cdots\!51}a^{14}+\frac{20\!\cdots\!35}{33\!\cdots\!93}a^{13}-\frac{69\!\cdots\!61}{48\!\cdots\!99}a^{12}+\frac{99\!\cdots\!50}{81\!\cdots\!93}a^{11}-\frac{36\!\cdots\!45}{11\!\cdots\!99}a^{10}+\frac{13\!\cdots\!19}{16\!\cdots\!57}a^{9}-\frac{10\!\cdots\!94}{23\!\cdots\!51}a^{8}-\frac{19\!\cdots\!18}{33\!\cdots\!93}a^{7}+\frac{74\!\cdots\!85}{48\!\cdots\!99}a^{6}-\frac{33\!\cdots\!27}{69\!\cdots\!57}a^{5}+\frac{57\!\cdots\!22}{99\!\cdots\!51}a^{4}-\frac{15\!\cdots\!54}{14\!\cdots\!93}a^{3}+\frac{23\!\cdots\!85}{28\!\cdots\!57}a^{2}-\frac{23\!\cdots\!39}{28\!\cdots\!57}a+\frac{61\!\cdots\!67}{41\!\cdots\!51}$, $\frac{12\!\cdots\!74}{11\!\cdots\!99}a^{35}+\frac{3394406136}{20\!\cdots\!99}a^{33}-\frac{128915822942237}{48\!\cdots\!99}a^{31}-\frac{48\!\cdots\!36}{11\!\cdots\!99}a^{29}-\frac{1021621479252}{20\!\cdots\!99}a^{27}+\frac{49\!\cdots\!13}{48\!\cdots\!99}a^{25}+\frac{14\!\cdots\!88}{11\!\cdots\!99}a^{23}+\frac{281226422586232}{20\!\cdots\!99}a^{21}-\frac{14\!\cdots\!97}{48\!\cdots\!99}a^{19}-\frac{17\!\cdots\!61}{11\!\cdots\!99}a^{17}-\frac{11\!\cdots\!32}{20\!\cdots\!99}a^{15}+\frac{16\!\cdots\!26}{48\!\cdots\!99}a^{13}+\frac{16\!\cdots\!11}{11\!\cdots\!99}a^{11}+\frac{758174666535132}{41\!\cdots\!51}a^{9}-\frac{15\!\cdots\!88}{48\!\cdots\!99}a^{7}-\frac{45\!\cdots\!49}{69\!\cdots\!57}a^{5}+\frac{33\!\cdots\!73}{20\!\cdots\!99}a^{3}-\frac{10\!\cdots\!37}{28\!\cdots\!57}a$, $\frac{29\!\cdots\!96}{11\!\cdots\!99}a^{34}-\frac{146689352}{54\!\cdots\!27}a^{32}-\frac{331206673974121}{48\!\cdots\!99}a^{30}-\frac{11\!\cdots\!10}{11\!\cdots\!99}a^{28}+\frac{44149399564}{54\!\cdots\!27}a^{26}+\frac{12\!\cdots\!29}{48\!\cdots\!99}a^{24}+\frac{35\!\cdots\!82}{11\!\cdots\!99}a^{22}-\frac{12515586075975}{54\!\cdots\!27}a^{20}-\frac{37\!\cdots\!22}{48\!\cdots\!99}a^{18}-\frac{44\!\cdots\!49}{11\!\cdots\!99}a^{16}+\frac{482916500836524}{54\!\cdots\!27}a^{14}+\frac{43\!\cdots\!37}{48\!\cdots\!99}a^{12}+\frac{40\!\cdots\!10}{11\!\cdots\!99}a^{10}-\frac{32764538502724}{11\!\cdots\!23}a^{8}-\frac{38\!\cdots\!59}{48\!\cdots\!99}a^{6}-\frac{30\!\cdots\!29}{99\!\cdots\!51}a^{4}-\frac{20\!\cdots\!44}{54\!\cdots\!27}a^{2}-\frac{17\!\cdots\!52}{41\!\cdots\!51}$
| 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: | \( 10470974244023312 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 10470974244023312 \cdot 15984}{36\cdot\sqrt{19575552194003861659818831697024491945650963873549328070183080689664}}\cr\approx \mathstrut & 0.244765769193906 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201)
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 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201, 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 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201);
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 - 394*x^30 + 118583*x^24 - 14205984*x^18 + 1297088703*x^12 - 4312188797*x^6 + 13841287201);
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 $C_6^2$ is not computed |
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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.1.0.1}{1} }^{36}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| Deg $36$ | $6$ | $6$ | $54$ | |||
|
\(19\)
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |