Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81)
gp: K = bnfinit(y^22 + 33*y^20 + 275*y^18 - 341*y^16 - 7062*y^14 - 1430*y^12 + 44902*y^10 + 9878*y^8 - 41371*y^6 + 10373*y^4 + 1463*y^2 - 81, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81)
\( x^{22} + 33 x^{20} + 275 x^{18} - 341 x^{16} - 7062 x^{14} - 1430 x^{12} + 44902 x^{10} + 9878 x^{8} + \cdots - 81 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(24689900716543842090569619202412690538496\)
\(\medspace = 2^{30}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(68.55\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{16}a^{3}+\frac{3}{16}a^{2}-\frac{1}{16}a-\frac{3}{16}$, $\frac{1}{16}a^{12}+\frac{1}{16}a^{8}+\frac{3}{16}a^{4}-\frac{5}{16}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{16}a+\frac{3}{8}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}+\frac{1}{32}a^{10}-\frac{1}{32}a^{8}+\frac{3}{32}a^{6}-\frac{3}{32}a^{4}-\frac{5}{32}a^{2}+\frac{5}{32}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{13}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{1}{16}a^{8}-\frac{1}{32}a^{7}-\frac{1}{8}a^{6}+\frac{1}{32}a^{5}+\frac{1}{8}a^{4}+\frac{1}{32}a^{3}+\frac{3}{16}a^{2}-\frac{1}{32}a-\frac{3}{16}$, $\frac{1}{64}a^{16}+\frac{1}{32}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a-\frac{11}{64}$, $\frac{1}{64}a^{17}+\frac{1}{32}a^{9}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{11}{64}a+\frac{1}{4}$, $\frac{1}{64}a^{18}+\frac{1}{32}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{5}{64}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{128}a^{19}-\frac{1}{128}a^{18}-\frac{1}{128}a^{17}-\frac{1}{128}a^{16}+\frac{1}{64}a^{11}-\frac{1}{64}a^{10}-\frac{1}{64}a^{9}+\frac{7}{64}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}-\frac{3}{16}a^{5}+\frac{1}{16}a^{4}+\frac{5}{128}a^{3}-\frac{5}{128}a^{2}+\frac{27}{128}a-\frac{21}{128}$, $\frac{1}{98\!\cdots\!76}a^{20}+\frac{32582458689181}{49\!\cdots\!88}a^{18}+\frac{37616276893101}{98\!\cdots\!76}a^{16}-\frac{13145115686425}{24\!\cdots\!44}a^{14}-\frac{108206329782801}{49\!\cdots\!88}a^{12}+\frac{9218912617097}{308521304573068}a^{10}+\frac{66313536820895}{49\!\cdots\!88}a^{8}+\frac{161021851392537}{24\!\cdots\!44}a^{6}-\frac{681933327832943}{98\!\cdots\!76}a^{4}-\frac{22\!\cdots\!25}{49\!\cdots\!88}a^{2}+\frac{35\!\cdots\!77}{98\!\cdots\!76}$, $\frac{1}{88\!\cdots\!84}a^{21}-\frac{106828904445991}{29\!\cdots\!28}a^{19}-\frac{1}{128}a^{18}+\frac{105816976902359}{22\!\cdots\!96}a^{17}-\frac{1}{128}a^{16}+\frac{141115536600109}{22\!\cdots\!96}a^{15}-\frac{36068776594267}{14\!\cdots\!64}a^{13}+\frac{378893580303353}{44\!\cdots\!92}a^{11}+\frac{3}{64}a^{10}-\frac{545320677664055}{22\!\cdots\!96}a^{9}+\frac{3}{64}a^{8}-\frac{147499453180531}{22\!\cdots\!96}a^{7}-\frac{1}{16}a^{6}-\frac{19\!\cdots\!59}{88\!\cdots\!84}a^{5}+\frac{3}{16}a^{4}-\frac{40\!\cdots\!81}{88\!\cdots\!84}a^{3}+\frac{3}{128}a^{2}+\frac{30\!\cdots\!71}{11\!\cdots\!48}a-\frac{29}{128}$
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
Trivial group, which has order $1$ (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: | $15$ | 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: |
$\frac{597927675311803}{88\!\cdots\!84}a^{21}+\frac{16\!\cdots\!35}{74\!\cdots\!32}a^{19}+\frac{16\!\cdots\!25}{88\!\cdots\!84}a^{17}-\frac{41\!\cdots\!41}{22\!\cdots\!96}a^{15}-\frac{70\!\cdots\!29}{14\!\cdots\!64}a^{13}-\frac{44\!\cdots\!47}{22\!\cdots\!96}a^{11}+\frac{13\!\cdots\!71}{44\!\cdots\!92}a^{9}+\frac{28\!\cdots\!97}{22\!\cdots\!96}a^{7}-\frac{20\!\cdots\!57}{88\!\cdots\!84}a^{5}+\frac{37\!\cdots\!87}{11\!\cdots\!48}a^{3}+\frac{63\!\cdots\!21}{88\!\cdots\!84}a$, $\frac{864340751876825}{88\!\cdots\!84}a^{21}+\frac{23\!\cdots\!05}{74\!\cdots\!32}a^{19}+\frac{24\!\cdots\!67}{88\!\cdots\!84}a^{17}-\frac{28\!\cdots\!71}{11\!\cdots\!48}a^{15}-\frac{10\!\cdots\!93}{14\!\cdots\!64}a^{13}-\frac{18\!\cdots\!25}{55\!\cdots\!24}a^{11}+\frac{18\!\cdots\!63}{44\!\cdots\!92}a^{9}+\frac{23\!\cdots\!59}{11\!\cdots\!48}a^{7}-\frac{28\!\cdots\!39}{88\!\cdots\!84}a^{5}+\frac{88\!\cdots\!01}{22\!\cdots\!96}a^{3}+\frac{10\!\cdots\!55}{88\!\cdots\!84}a$, $\frac{102445104442581}{98\!\cdots\!76}a^{20}+\frac{17\!\cdots\!07}{49\!\cdots\!88}a^{18}+\frac{29\!\cdots\!97}{98\!\cdots\!76}a^{16}-\frac{25\!\cdots\!73}{12\!\cdots\!72}a^{14}-\frac{36\!\cdots\!23}{49\!\cdots\!88}a^{12}-\frac{12\!\cdots\!11}{24\!\cdots\!44}a^{10}+\frac{21\!\cdots\!33}{49\!\cdots\!88}a^{8}+\frac{39\!\cdots\!35}{12\!\cdots\!72}a^{6}-\frac{27\!\cdots\!31}{98\!\cdots\!76}a^{4}-\frac{68\!\cdots\!17}{49\!\cdots\!88}a^{2}+\frac{89\!\cdots\!17}{98\!\cdots\!76}$, $\frac{23\!\cdots\!23}{88\!\cdots\!84}a^{21}+\frac{16\!\cdots\!47}{18\!\cdots\!08}a^{19}+\frac{65\!\cdots\!13}{88\!\cdots\!84}a^{17}-\frac{20\!\cdots\!05}{22\!\cdots\!96}a^{15}-\frac{27\!\cdots\!37}{14\!\cdots\!64}a^{13}-\frac{89\!\cdots\!33}{22\!\cdots\!96}a^{11}+\frac{53\!\cdots\!23}{44\!\cdots\!92}a^{9}+\frac{62\!\cdots\!01}{22\!\cdots\!96}a^{7}-\frac{99\!\cdots\!89}{88\!\cdots\!84}a^{5}+\frac{54\!\cdots\!61}{22\!\cdots\!96}a^{3}+\frac{45\!\cdots\!73}{88\!\cdots\!84}a$, $\frac{14\!\cdots\!97}{88\!\cdots\!84}a^{21}+\frac{39\!\cdots\!23}{74\!\cdots\!32}a^{19}+\frac{39\!\cdots\!11}{88\!\cdots\!84}a^{17}-\frac{11\!\cdots\!81}{22\!\cdots\!96}a^{15}-\frac{16\!\cdots\!83}{14\!\cdots\!64}a^{13}-\frac{59\!\cdots\!15}{22\!\cdots\!96}a^{11}+\frac{32\!\cdots\!73}{44\!\cdots\!92}a^{9}+\frac{40\!\cdots\!77}{22\!\cdots\!96}a^{7}-\frac{59\!\cdots\!67}{88\!\cdots\!84}a^{5}+\frac{16\!\cdots\!75}{11\!\cdots\!48}a^{3}+\frac{27\!\cdots\!67}{88\!\cdots\!84}a$, $\frac{12642067136229}{24\!\cdots\!44}a^{20}+\frac{825550672299981}{49\!\cdots\!88}a^{18}+\frac{66\!\cdots\!49}{49\!\cdots\!88}a^{16}-\frac{13\!\cdots\!57}{617042609146136}a^{14}-\frac{44\!\cdots\!55}{12\!\cdots\!72}a^{12}+\frac{13\!\cdots\!57}{24\!\cdots\!44}a^{10}+\frac{58\!\cdots\!69}{24\!\cdots\!44}a^{8}-\frac{41\!\cdots\!73}{154260652286534}a^{6}-\frac{60\!\cdots\!23}{24\!\cdots\!44}a^{4}+\frac{54\!\cdots\!57}{49\!\cdots\!88}a^{2}-\frac{98\!\cdots\!95}{49\!\cdots\!88}$, $\frac{97795697977879}{88\!\cdots\!84}a^{21}+\frac{550807554105475}{14\!\cdots\!64}a^{19}+\frac{29\!\cdots\!11}{88\!\cdots\!84}a^{17}-\frac{82278130798063}{694172935289403}a^{15}-\frac{11\!\cdots\!03}{14\!\cdots\!64}a^{13}-\frac{17\!\cdots\!23}{22\!\cdots\!96}a^{11}+\frac{19\!\cdots\!39}{44\!\cdots\!92}a^{9}+\frac{26\!\cdots\!39}{55\!\cdots\!24}a^{7}-\frac{14\!\cdots\!21}{88\!\cdots\!84}a^{5}-\frac{34\!\cdots\!55}{44\!\cdots\!92}a^{3}+\frac{77\!\cdots\!19}{88\!\cdots\!84}a$, $\frac{188367915517661}{88\!\cdots\!84}a^{21}+\frac{276369331246925}{37\!\cdots\!16}a^{19}+\frac{65\!\cdots\!67}{88\!\cdots\!84}a^{17}+\frac{36\!\cdots\!81}{55\!\cdots\!24}a^{15}-\frac{23\!\cdots\!21}{14\!\cdots\!64}a^{13}-\frac{20\!\cdots\!07}{55\!\cdots\!24}a^{11}+\frac{30\!\cdots\!03}{44\!\cdots\!92}a^{9}+\frac{12\!\cdots\!81}{55\!\cdots\!24}a^{7}+\frac{64\!\cdots\!53}{88\!\cdots\!84}a^{5}-\frac{90\!\cdots\!05}{11\!\cdots\!48}a^{3}-\frac{89\!\cdots\!93}{88\!\cdots\!84}a$, $\frac{184084252015663}{11\!\cdots\!48}a^{21}+\frac{10\!\cdots\!95}{18\!\cdots\!08}a^{19}+\frac{32\!\cdots\!32}{694172935289403}a^{17}-\frac{89\!\cdots\!73}{22\!\cdots\!96}a^{15}-\frac{87\!\cdots\!47}{74\!\cdots\!32}a^{13}-\frac{14\!\cdots\!53}{22\!\cdots\!96}a^{11}+\frac{15\!\cdots\!39}{22\!\cdots\!96}a^{9}+\frac{89\!\cdots\!37}{22\!\cdots\!96}a^{7}-\frac{11\!\cdots\!01}{22\!\cdots\!96}a^{5}+\frac{67\!\cdots\!45}{22\!\cdots\!96}a^{3}+\frac{33\!\cdots\!69}{22\!\cdots\!96}a$, $\frac{42063116334965}{49\!\cdots\!88}a^{20}+\frac{14\!\cdots\!21}{49\!\cdots\!88}a^{18}+\frac{384156362351545}{154260652286534}a^{16}-\frac{999566330011319}{617042609146136}a^{14}-\frac{15\!\cdots\!37}{24\!\cdots\!44}a^{12}-\frac{10\!\cdots\!75}{24\!\cdots\!44}a^{10}+\frac{44\!\cdots\!07}{12\!\cdots\!72}a^{8}+\frac{82\!\cdots\!99}{308521304573068}a^{6}-\frac{10\!\cdots\!51}{49\!\cdots\!88}a^{4}-\frac{82\!\cdots\!99}{49\!\cdots\!88}a^{2}-\frac{26\!\cdots\!05}{12\!\cdots\!72}$, $\frac{719302542207953}{22\!\cdots\!96}a^{21}-\frac{113560218323043}{49\!\cdots\!88}a^{20}+\frac{40\!\cdots\!27}{37\!\cdots\!16}a^{19}-\frac{38\!\cdots\!11}{49\!\cdots\!88}a^{18}+\frac{10\!\cdots\!51}{11\!\cdots\!48}a^{17}-\frac{83\!\cdots\!99}{12\!\cdots\!72}a^{16}-\frac{12\!\cdots\!93}{22\!\cdots\!96}a^{15}+\frac{95\!\cdots\!97}{24\!\cdots\!44}a^{14}-\frac{17\!\cdots\!07}{74\!\cdots\!32}a^{13}+\frac{20\!\cdots\!17}{12\!\cdots\!72}a^{12}-\frac{38\!\cdots\!07}{22\!\cdots\!96}a^{11}+\frac{80\!\cdots\!25}{617042609146136}a^{10}+\frac{30\!\cdots\!11}{22\!\cdots\!96}a^{9}-\frac{23\!\cdots\!29}{24\!\cdots\!44}a^{8}+\frac{23\!\cdots\!25}{22\!\cdots\!96}a^{7}-\frac{19\!\cdots\!65}{24\!\cdots\!44}a^{6}-\frac{10\!\cdots\!63}{13\!\cdots\!06}a^{5}+\frac{24\!\cdots\!87}{49\!\cdots\!88}a^{4}-\frac{14\!\cdots\!59}{22\!\cdots\!96}a^{3}+\frac{36\!\cdots\!83}{49\!\cdots\!88}a^{2}+\frac{26\!\cdots\!27}{22\!\cdots\!96}a-\frac{82\!\cdots\!81}{24\!\cdots\!44}$, $\frac{782544386424661}{29\!\cdots\!28}a^{21}-\frac{201158101324499}{98\!\cdots\!76}a^{20}+\frac{10\!\cdots\!51}{12\!\cdots\!72}a^{19}-\frac{33\!\cdots\!85}{49\!\cdots\!88}a^{18}+\frac{22\!\cdots\!57}{29\!\cdots\!28}a^{17}-\frac{58\!\cdots\!37}{98\!\cdots\!76}a^{16}-\frac{17\!\cdots\!37}{37\!\cdots\!16}a^{15}+\frac{93\!\cdots\!81}{24\!\cdots\!44}a^{14}-\frac{93\!\cdots\!89}{49\!\cdots\!88}a^{13}+\frac{72\!\cdots\!79}{49\!\cdots\!88}a^{12}-\frac{13\!\cdots\!35}{925563913719204}a^{11}+\frac{13\!\cdots\!37}{12\!\cdots\!72}a^{10}+\frac{16\!\cdots\!97}{14\!\cdots\!64}a^{9}-\frac{42\!\cdots\!95}{49\!\cdots\!88}a^{8}+\frac{32\!\cdots\!67}{37\!\cdots\!16}a^{7}-\frac{16\!\cdots\!09}{24\!\cdots\!44}a^{6}-\frac{18\!\cdots\!07}{29\!\cdots\!28}a^{5}+\frac{48\!\cdots\!01}{98\!\cdots\!76}a^{4}-\frac{25\!\cdots\!39}{37\!\cdots\!16}a^{3}+\frac{22\!\cdots\!13}{49\!\cdots\!88}a^{2}+\frac{26\!\cdots\!81}{29\!\cdots\!28}a-\frac{35\!\cdots\!13}{98\!\cdots\!76}$, $\frac{15774306196304}{694172935289403}a^{21}-\frac{83249540887649}{98\!\cdots\!76}a^{20}+\frac{55\!\cdots\!01}{74\!\cdots\!32}a^{19}-\frac{13\!\cdots\!45}{49\!\cdots\!88}a^{18}+\frac{27\!\cdots\!55}{44\!\cdots\!92}a^{17}-\frac{23\!\cdots\!21}{98\!\cdots\!76}a^{16}-\frac{41\!\cdots\!31}{55\!\cdots\!24}a^{15}+\frac{49\!\cdots\!47}{24\!\cdots\!44}a^{14}-\frac{37\!\cdots\!05}{231390978429801}a^{13}+\frac{29\!\cdots\!05}{49\!\cdots\!88}a^{12}-\frac{11\!\cdots\!93}{27\!\cdots\!12}a^{11}+\frac{10\!\cdots\!27}{308521304573068}a^{10}+\frac{22\!\cdots\!65}{22\!\cdots\!96}a^{9}-\frac{17\!\cdots\!71}{49\!\cdots\!88}a^{8}+\frac{19\!\cdots\!44}{694172935289403}a^{7}-\frac{51\!\cdots\!87}{24\!\cdots\!44}a^{6}-\frac{12\!\cdots\!19}{13\!\cdots\!06}a^{5}+\frac{25\!\cdots\!87}{98\!\cdots\!76}a^{4}+\frac{41\!\cdots\!25}{22\!\cdots\!96}a^{3}-\frac{54\!\cdots\!15}{49\!\cdots\!88}a^{2}+\frac{18\!\cdots\!19}{44\!\cdots\!92}a-\frac{88\!\cdots\!37}{98\!\cdots\!76}$, $\frac{73\!\cdots\!23}{88\!\cdots\!84}a^{21}-\frac{86303667288989}{24\!\cdots\!44}a^{20}+\frac{20\!\cdots\!57}{74\!\cdots\!32}a^{19}-\frac{721188784477747}{617042609146136}a^{18}+\frac{20\!\cdots\!13}{88\!\cdots\!84}a^{17}-\frac{49\!\cdots\!79}{49\!\cdots\!88}a^{16}-\frac{24\!\cdots\!29}{11\!\cdots\!48}a^{15}+\frac{18\!\cdots\!19}{24\!\cdots\!44}a^{14}-\frac{86\!\cdots\!91}{14\!\cdots\!64}a^{13}+\frac{61\!\cdots\!83}{24\!\cdots\!44}a^{12}-\frac{30\!\cdots\!79}{11\!\cdots\!48}a^{11}+\frac{38\!\cdots\!13}{24\!\cdots\!44}a^{10}+\frac{15\!\cdots\!37}{44\!\cdots\!92}a^{9}-\frac{46\!\cdots\!93}{308521304573068}a^{8}+\frac{19\!\cdots\!75}{11\!\cdots\!48}a^{7}-\frac{23\!\cdots\!99}{24\!\cdots\!44}a^{6}-\frac{24\!\cdots\!53}{88\!\cdots\!84}a^{5}+\frac{11\!\cdots\!77}{12\!\cdots\!72}a^{4}+\frac{72\!\cdots\!79}{22\!\cdots\!96}a^{3}-\frac{33\!\cdots\!69}{24\!\cdots\!44}a^{2}+\frac{80\!\cdots\!97}{88\!\cdots\!84}a-\frac{10\!\cdots\!45}{49\!\cdots\!88}$, $\frac{73\!\cdots\!57}{88\!\cdots\!84}a^{21}+\frac{675500181746605}{98\!\cdots\!76}a^{20}+\frac{82\!\cdots\!31}{29\!\cdots\!28}a^{19}+\frac{22\!\cdots\!67}{98\!\cdots\!76}a^{18}+\frac{10\!\cdots\!77}{44\!\cdots\!92}a^{17}+\frac{61\!\cdots\!13}{308521304573068}a^{16}-\frac{17\!\cdots\!83}{11\!\cdots\!48}a^{15}-\frac{32\!\cdots\!53}{24\!\cdots\!44}a^{14}-\frac{88\!\cdots\!57}{14\!\cdots\!64}a^{13}-\frac{24\!\cdots\!73}{49\!\cdots\!88}a^{12}-\frac{18\!\cdots\!75}{44\!\cdots\!92}a^{11}-\frac{17\!\cdots\!31}{49\!\cdots\!88}a^{10}+\frac{77\!\cdots\!63}{22\!\cdots\!96}a^{9}+\frac{71\!\cdots\!99}{24\!\cdots\!44}a^{8}+\frac{14\!\cdots\!55}{55\!\cdots\!24}a^{7}+\frac{53\!\cdots\!91}{24\!\cdots\!44}a^{6}-\frac{18\!\cdots\!47}{88\!\cdots\!84}a^{5}-\frac{16\!\cdots\!43}{98\!\cdots\!76}a^{4}-\frac{19\!\cdots\!51}{88\!\cdots\!84}a^{3}-\frac{16\!\cdots\!09}{98\!\cdots\!76}a^{2}+\frac{54\!\cdots\!61}{44\!\cdots\!92}a+\frac{26\!\cdots\!19}{24\!\cdots\!44}$
| 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: | \( 6858150642190 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 6858150642190 \cdot 1}{2\cdot\sqrt{24689900716543842090569619202412690538496}}\cr\approx \mathstrut & 1.37498043877620 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81)
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^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81, 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^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81);
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^22 + 33*x^20 + 275*x^18 - 341*x^16 - 7062*x^14 - 1430*x^12 + 44902*x^10 + 9878*x^8 - 41371*x^6 + 10373*x^4 + 1463*x^2 - 81);
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
$C_2^{10}.F_{11}$ (as 22T34):
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 solvable group of order 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ |
Intermediate fields
| 11.11.4910318845910094848.1 |
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]
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $22$ | $1$ | $30$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.20.10.1 | $x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |