Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900)
gp: K = bnfinit(y^21 - 21*y^17 - 280*y^15 - 138*y^14 + 98*y^13 + 1792*y^12 + 2352*y^11 + 1008*y^10 + 13279*y^9 - 44576*y^8 + 4321*y^7 + 142296*y^6 - 106484*y^5 + 32928*y^4 + 355376*y^3 - 154252*y^2 - 254912*y + 83900, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900)
\( x^{21} - 21 x^{17} - 280 x^{15} - 138 x^{14} + 98 x^{13} + 1792 x^{12} + 2352 x^{11} + 1008 x^{10} + \cdots + 83900 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[7, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-231636832837360510908018449295052404602048\)
\(\medspace = -\,2^{6}\cdot 7^{30}\cdot 107^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(93.27\) | 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: |
\(2\), \(7\), \(107\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-107}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{2122}a^{18}-\frac{101}{2122}a^{17}-\frac{83}{1061}a^{16}-\frac{140}{1061}a^{15}-\frac{51}{2122}a^{14}-\frac{239}{2122}a^{13}+\frac{177}{1061}a^{12}+\frac{73}{2122}a^{11}+\frac{479}{2122}a^{10}-\frac{885}{2122}a^{9}-\frac{17}{2122}a^{8}-\frac{276}{1061}a^{7}-\frac{233}{2122}a^{6}-\frac{201}{1061}a^{5}-\frac{469}{2122}a^{4}+\frac{169}{1061}a^{3}+\frac{279}{1061}a^{2}+\frac{205}{1061}a+\frac{7}{1061}$, $\frac{1}{2122}a^{19}+\frac{243}{2122}a^{17}-\frac{35}{1061}a^{16}+\frac{158}{1061}a^{15}-\frac{85}{2122}a^{14}-\frac{443}{2122}a^{13}-\frac{247}{2122}a^{12}+\frac{425}{2122}a^{11}-\frac{251}{2122}a^{10}+\frac{783}{2122}a^{9}-\frac{147}{2122}a^{8}-\frac{813}{2122}a^{7}-\frac{593}{2122}a^{6}+\frac{154}{1061}a^{5}-\frac{347}{2122}a^{4}+\frac{372}{1061}a^{3}-\frac{263}{1061}a^{2}-\frac{508}{1061}a-\frac{354}{1061}$, $\frac{1}{21\!\cdots\!76}a^{20}-\frac{22\!\cdots\!67}{21\!\cdots\!76}a^{19}+\frac{33\!\cdots\!41}{21\!\cdots\!76}a^{18}-\frac{15\!\cdots\!87}{21\!\cdots\!76}a^{17}+\frac{83\!\cdots\!83}{52\!\cdots\!44}a^{16}-\frac{10\!\cdots\!27}{52\!\cdots\!44}a^{15}+\frac{12\!\cdots\!25}{52\!\cdots\!44}a^{14}+\frac{22\!\cdots\!37}{10\!\cdots\!88}a^{13}+\frac{11\!\cdots\!07}{52\!\cdots\!44}a^{12}-\frac{10\!\cdots\!43}{52\!\cdots\!44}a^{11}+\frac{16\!\cdots\!07}{52\!\cdots\!44}a^{10}-\frac{10\!\cdots\!65}{31\!\cdots\!32}a^{9}-\frac{14\!\cdots\!93}{21\!\cdots\!76}a^{8}-\frac{42\!\cdots\!17}{21\!\cdots\!76}a^{7}-\frac{19\!\cdots\!17}{52\!\cdots\!44}a^{6}-\frac{15\!\cdots\!89}{52\!\cdots\!44}a^{5}-\frac{51\!\cdots\!83}{26\!\cdots\!22}a^{4}+\frac{12\!\cdots\!08}{13\!\cdots\!11}a^{3}-\frac{87\!\cdots\!61}{13\!\cdots\!11}a^{2}+\frac{11\!\cdots\!29}{52\!\cdots\!44}a+\frac{23\!\cdots\!05}{52\!\cdots\!44}$
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: | $13$ | 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: |
$\frac{15\!\cdots\!95}{19\!\cdots\!16}a^{20}-\frac{13\!\cdots\!29}{19\!\cdots\!16}a^{19}+\frac{11\!\cdots\!99}{19\!\cdots\!16}a^{18}-\frac{14\!\cdots\!09}{19\!\cdots\!16}a^{17}-\frac{79\!\cdots\!15}{49\!\cdots\!04}a^{16}+\frac{71\!\cdots\!33}{49\!\cdots\!04}a^{15}-\frac{11\!\cdots\!87}{49\!\cdots\!04}a^{14}+\frac{99\!\cdots\!35}{99\!\cdots\!08}a^{13}+\frac{30\!\cdots\!51}{49\!\cdots\!04}a^{12}+\frac{70\!\cdots\!91}{49\!\cdots\!04}a^{11}+\frac{38\!\cdots\!95}{49\!\cdots\!04}a^{10}-\frac{42\!\cdots\!17}{29\!\cdots\!12}a^{9}+\frac{18\!\cdots\!65}{19\!\cdots\!16}a^{8}-\frac{89\!\cdots\!51}{19\!\cdots\!16}a^{7}+\frac{19\!\cdots\!17}{49\!\cdots\!04}a^{6}+\frac{36\!\cdots\!31}{49\!\cdots\!04}a^{5}-\frac{16\!\cdots\!86}{12\!\cdots\!51}a^{4}+\frac{23\!\cdots\!40}{12\!\cdots\!51}a^{3}+\frac{10\!\cdots\!97}{12\!\cdots\!51}a^{2}-\frac{12\!\cdots\!25}{49\!\cdots\!04}a+\frac{46\!\cdots\!67}{49\!\cdots\!04}$, $\frac{24\!\cdots\!87}{10\!\cdots\!88}a^{20}-\frac{29\!\cdots\!49}{10\!\cdots\!88}a^{19}+\frac{32\!\cdots\!07}{10\!\cdots\!88}a^{18}-\frac{54\!\cdots\!33}{10\!\cdots\!88}a^{17}-\frac{11\!\cdots\!23}{26\!\cdots\!22}a^{16}+\frac{60\!\cdots\!66}{13\!\cdots\!11}a^{15}-\frac{92\!\cdots\!29}{13\!\cdots\!11}a^{14}+\frac{29\!\cdots\!57}{52\!\cdots\!44}a^{13}-\frac{83\!\cdots\!67}{26\!\cdots\!22}a^{12}+\frac{67\!\cdots\!26}{13\!\cdots\!11}a^{11}+\frac{57\!\cdots\!99}{26\!\cdots\!22}a^{10}+\frac{30\!\cdots\!39}{77\!\cdots\!83}a^{9}+\frac{24\!\cdots\!57}{10\!\cdots\!88}a^{8}-\frac{14\!\cdots\!03}{10\!\cdots\!88}a^{7}+\frac{39\!\cdots\!81}{26\!\cdots\!22}a^{6}+\frac{33\!\cdots\!05}{26\!\cdots\!22}a^{5}-\frac{88\!\cdots\!27}{26\!\cdots\!22}a^{4}+\frac{64\!\cdots\!06}{13\!\cdots\!11}a^{3}+\frac{50\!\cdots\!04}{13\!\cdots\!11}a^{2}-\frac{24\!\cdots\!31}{26\!\cdots\!22}a-\frac{14\!\cdots\!79}{26\!\cdots\!22}$, $\frac{36\!\cdots\!95}{10\!\cdots\!88}a^{20}+\frac{30\!\cdots\!87}{10\!\cdots\!88}a^{19}+\frac{22\!\cdots\!19}{10\!\cdots\!88}a^{18}+\frac{25\!\cdots\!15}{10\!\cdots\!88}a^{17}-\frac{93\!\cdots\!70}{13\!\cdots\!11}a^{16}-\frac{15\!\cdots\!39}{26\!\cdots\!22}a^{15}-\frac{13\!\cdots\!90}{13\!\cdots\!11}a^{14}-\frac{69\!\cdots\!37}{52\!\cdots\!44}a^{13}-\frac{90\!\cdots\!58}{13\!\cdots\!11}a^{12}+\frac{71\!\cdots\!75}{13\!\cdots\!11}a^{11}+\frac{33\!\cdots\!01}{26\!\cdots\!22}a^{10}+\frac{10\!\cdots\!21}{77\!\cdots\!83}a^{9}+\frac{59\!\cdots\!25}{10\!\cdots\!88}a^{8}-\frac{11\!\cdots\!15}{10\!\cdots\!88}a^{7}-\frac{99\!\cdots\!93}{13\!\cdots\!11}a^{6}+\frac{60\!\cdots\!45}{13\!\cdots\!11}a^{5}-\frac{67\!\cdots\!59}{26\!\cdots\!22}a^{4}+\frac{18\!\cdots\!78}{13\!\cdots\!11}a^{3}+\frac{17\!\cdots\!98}{13\!\cdots\!11}a^{2}+\frac{12\!\cdots\!21}{26\!\cdots\!22}a-\frac{17\!\cdots\!39}{26\!\cdots\!22}$, $\frac{79\!\cdots\!55}{21\!\cdots\!76}a^{20}+\frac{14\!\cdots\!11}{21\!\cdots\!76}a^{19}+\frac{51\!\cdots\!71}{21\!\cdots\!76}a^{18}+\frac{11\!\cdots\!71}{21\!\cdots\!76}a^{17}-\frac{40\!\cdots\!43}{52\!\cdots\!44}a^{16}-\frac{76\!\cdots\!27}{52\!\cdots\!44}a^{15}-\frac{58\!\cdots\!53}{52\!\cdots\!44}a^{14}-\frac{27\!\cdots\!57}{10\!\cdots\!88}a^{13}-\frac{70\!\cdots\!33}{52\!\cdots\!44}a^{12}+\frac{29\!\cdots\!61}{52\!\cdots\!44}a^{11}+\frac{10\!\cdots\!01}{52\!\cdots\!44}a^{10}+\frac{74\!\cdots\!73}{31\!\cdots\!32}a^{9}+\frac{15\!\cdots\!41}{21\!\cdots\!76}a^{8}-\frac{11\!\cdots\!47}{21\!\cdots\!76}a^{7}-\frac{13\!\cdots\!65}{52\!\cdots\!44}a^{6}+\frac{28\!\cdots\!07}{52\!\cdots\!44}a^{5}+\frac{11\!\cdots\!95}{26\!\cdots\!22}a^{4}-\frac{41\!\cdots\!73}{13\!\cdots\!11}a^{3}+\frac{26\!\cdots\!41}{13\!\cdots\!11}a^{2}+\frac{88\!\cdots\!47}{52\!\cdots\!44}a-\frac{44\!\cdots\!49}{52\!\cdots\!44}$, $\frac{24\!\cdots\!41}{52\!\cdots\!44}a^{20}+\frac{17\!\cdots\!45}{52\!\cdots\!44}a^{19}+\frac{42\!\cdots\!97}{52\!\cdots\!44}a^{18}-\frac{33\!\cdots\!63}{52\!\cdots\!44}a^{17}-\frac{10\!\cdots\!98}{13\!\cdots\!11}a^{16}-\frac{10\!\cdots\!02}{13\!\cdots\!11}a^{15}-\frac{37\!\cdots\!39}{26\!\cdots\!22}a^{14}-\frac{38\!\cdots\!41}{26\!\cdots\!22}a^{13}-\frac{33\!\cdots\!38}{13\!\cdots\!11}a^{12}+\frac{12\!\cdots\!54}{13\!\cdots\!11}a^{11}+\frac{35\!\cdots\!31}{26\!\cdots\!22}a^{10}+\frac{21\!\cdots\!79}{77\!\cdots\!83}a^{9}+\frac{35\!\cdots\!49}{52\!\cdots\!44}a^{8}-\frac{73\!\cdots\!99}{52\!\cdots\!44}a^{7}-\frac{28\!\cdots\!77}{26\!\cdots\!22}a^{6}+\frac{70\!\cdots\!07}{26\!\cdots\!22}a^{5}+\frac{65\!\cdots\!37}{13\!\cdots\!11}a^{4}+\frac{11\!\cdots\!39}{13\!\cdots\!11}a^{3}+\frac{11\!\cdots\!97}{13\!\cdots\!11}a^{2}+\frac{26\!\cdots\!98}{13\!\cdots\!11}a-\frac{99\!\cdots\!02}{13\!\cdots\!11}$, $\frac{61\!\cdots\!61}{21\!\cdots\!76}a^{20}+\frac{38\!\cdots\!49}{21\!\cdots\!76}a^{19}-\frac{26\!\cdots\!63}{21\!\cdots\!76}a^{18}+\frac{45\!\cdots\!93}{21\!\cdots\!76}a^{17}-\frac{32\!\cdots\!15}{52\!\cdots\!44}a^{16}-\frac{18\!\cdots\!23}{52\!\cdots\!44}a^{15}-\frac{41\!\cdots\!71}{52\!\cdots\!44}a^{14}-\frac{10\!\cdots\!63}{10\!\cdots\!88}a^{13}+\frac{21\!\cdots\!39}{52\!\cdots\!44}a^{12}+\frac{25\!\cdots\!33}{52\!\cdots\!44}a^{11}+\frac{53\!\cdots\!39}{52\!\cdots\!44}a^{10}+\frac{13\!\cdots\!83}{31\!\cdots\!32}a^{9}+\frac{89\!\cdots\!31}{21\!\cdots\!76}a^{8}-\frac{22\!\cdots\!25}{21\!\cdots\!76}a^{7}-\frac{41\!\cdots\!25}{52\!\cdots\!44}a^{6}+\frac{26\!\cdots\!67}{52\!\cdots\!44}a^{5}-\frac{22\!\cdots\!08}{13\!\cdots\!11}a^{4}-\frac{22\!\cdots\!80}{13\!\cdots\!11}a^{3}+\frac{17\!\cdots\!92}{13\!\cdots\!11}a^{2}+\frac{19\!\cdots\!21}{52\!\cdots\!44}a-\frac{54\!\cdots\!99}{52\!\cdots\!44}$, $\frac{35\!\cdots\!83}{21\!\cdots\!76}a^{20}+\frac{38\!\cdots\!35}{21\!\cdots\!76}a^{19}+\frac{12\!\cdots\!87}{21\!\cdots\!76}a^{18}-\frac{13\!\cdots\!57}{21\!\cdots\!76}a^{17}-\frac{18\!\cdots\!75}{52\!\cdots\!44}a^{16}-\frac{30\!\cdots\!39}{52\!\cdots\!44}a^{15}-\frac{25\!\cdots\!35}{52\!\cdots\!44}a^{14}-\frac{29\!\cdots\!93}{10\!\cdots\!88}a^{13}-\frac{18\!\cdots\!03}{52\!\cdots\!44}a^{12}+\frac{16\!\cdots\!19}{52\!\cdots\!44}a^{11}+\frac{21\!\cdots\!07}{52\!\cdots\!44}a^{10}+\frac{11\!\cdots\!45}{31\!\cdots\!32}a^{9}+\frac{47\!\cdots\!29}{21\!\cdots\!76}a^{8}-\frac{15\!\cdots\!95}{21\!\cdots\!76}a^{7}+\frac{21\!\cdots\!11}{52\!\cdots\!44}a^{6}+\frac{11\!\cdots\!13}{52\!\cdots\!44}a^{5}-\frac{31\!\cdots\!99}{26\!\cdots\!22}a^{4}+\frac{84\!\cdots\!32}{13\!\cdots\!11}a^{3}+\frac{71\!\cdots\!77}{13\!\cdots\!11}a^{2}-\frac{46\!\cdots\!17}{52\!\cdots\!44}a-\frac{21\!\cdots\!41}{52\!\cdots\!44}$, $\frac{50\!\cdots\!09}{21\!\cdots\!76}a^{20}-\frac{77\!\cdots\!35}{21\!\cdots\!76}a^{19}+\frac{91\!\cdots\!81}{21\!\cdots\!76}a^{18}-\frac{11\!\cdots\!75}{21\!\cdots\!76}a^{17}-\frac{23\!\cdots\!87}{52\!\cdots\!44}a^{16}+\frac{37\!\cdots\!93}{52\!\cdots\!44}a^{15}-\frac{39\!\cdots\!17}{52\!\cdots\!44}a^{14}+\frac{85\!\cdots\!65}{10\!\cdots\!88}a^{13}-\frac{30\!\cdots\!35}{52\!\cdots\!44}a^{12}+\frac{25\!\cdots\!51}{52\!\cdots\!44}a^{11}-\frac{82\!\cdots\!75}{52\!\cdots\!44}a^{10}+\frac{56\!\cdots\!45}{31\!\cdots\!32}a^{9}+\frac{58\!\cdots\!39}{21\!\cdots\!76}a^{8}-\frac{32\!\cdots\!49}{21\!\cdots\!76}a^{7}+\frac{12\!\cdots\!35}{52\!\cdots\!44}a^{6}+\frac{30\!\cdots\!47}{52\!\cdots\!44}a^{5}-\frac{58\!\cdots\!93}{13\!\cdots\!11}a^{4}+\frac{98\!\cdots\!20}{13\!\cdots\!11}a^{3}-\frac{16\!\cdots\!73}{13\!\cdots\!11}a^{2}-\frac{25\!\cdots\!31}{52\!\cdots\!44}a+\frac{10\!\cdots\!49}{52\!\cdots\!44}$, $\frac{41\!\cdots\!89}{21\!\cdots\!76}a^{20}+\frac{75\!\cdots\!49}{21\!\cdots\!76}a^{19}-\frac{31\!\cdots\!55}{21\!\cdots\!76}a^{18}-\frac{10\!\cdots\!19}{21\!\cdots\!76}a^{17}-\frac{29\!\cdots\!25}{52\!\cdots\!44}a^{16}-\frac{15\!\cdots\!09}{52\!\cdots\!44}a^{15}-\frac{30\!\cdots\!89}{52\!\cdots\!44}a^{14}-\frac{36\!\cdots\!23}{10\!\cdots\!88}a^{13}+\frac{36\!\cdots\!83}{52\!\cdots\!44}a^{12}+\frac{28\!\cdots\!27}{52\!\cdots\!44}a^{11}+\frac{53\!\cdots\!47}{52\!\cdots\!44}a^{10}+\frac{23\!\cdots\!39}{31\!\cdots\!32}a^{9}+\frac{60\!\cdots\!91}{21\!\cdots\!76}a^{8}-\frac{21\!\cdots\!89}{21\!\cdots\!76}a^{7}-\frac{30\!\cdots\!83}{52\!\cdots\!44}a^{6}+\frac{11\!\cdots\!37}{52\!\cdots\!44}a^{5}-\frac{44\!\cdots\!85}{26\!\cdots\!22}a^{4}+\frac{16\!\cdots\!23}{13\!\cdots\!11}a^{3}+\frac{97\!\cdots\!17}{13\!\cdots\!11}a^{2}-\frac{54\!\cdots\!19}{52\!\cdots\!44}a-\frac{21\!\cdots\!87}{52\!\cdots\!44}$, $\frac{11\!\cdots\!07}{10\!\cdots\!88}a^{20}-\frac{53\!\cdots\!49}{10\!\cdots\!88}a^{19}+\frac{10\!\cdots\!51}{10\!\cdots\!88}a^{18}-\frac{13\!\cdots\!29}{10\!\cdots\!88}a^{17}-\frac{53\!\cdots\!87}{26\!\cdots\!22}a^{16}+\frac{10\!\cdots\!51}{13\!\cdots\!11}a^{15}-\frac{77\!\cdots\!43}{24\!\cdots\!02}a^{14}+\frac{10\!\cdots\!11}{52\!\cdots\!44}a^{13}-\frac{37\!\cdots\!73}{26\!\cdots\!22}a^{12}+\frac{27\!\cdots\!55}{13\!\cdots\!11}a^{11}+\frac{33\!\cdots\!79}{26\!\cdots\!22}a^{10}+\frac{32\!\cdots\!01}{15\!\cdots\!66}a^{9}+\frac{13\!\cdots\!57}{10\!\cdots\!88}a^{8}-\frac{54\!\cdots\!23}{10\!\cdots\!88}a^{7}+\frac{98\!\cdots\!57}{26\!\cdots\!22}a^{6}+\frac{11\!\cdots\!37}{13\!\cdots\!11}a^{5}-\frac{26\!\cdots\!39}{26\!\cdots\!22}a^{4}+\frac{15\!\cdots\!75}{13\!\cdots\!11}a^{3}+\frac{28\!\cdots\!95}{13\!\cdots\!11}a^{2}-\frac{31\!\cdots\!05}{26\!\cdots\!22}a-\frac{34\!\cdots\!83}{26\!\cdots\!22}$, $\frac{18\!\cdots\!71}{52\!\cdots\!44}a^{20}-\frac{32\!\cdots\!21}{52\!\cdots\!44}a^{19}+\frac{24\!\cdots\!71}{52\!\cdots\!44}a^{18}-\frac{79\!\cdots\!73}{52\!\cdots\!44}a^{17}-\frac{14\!\cdots\!35}{26\!\cdots\!22}a^{16}-\frac{45\!\cdots\!79}{26\!\cdots\!22}a^{15}-\frac{26\!\cdots\!73}{26\!\cdots\!22}a^{14}-\frac{17\!\cdots\!10}{13\!\cdots\!11}a^{13}-\frac{13\!\cdots\!92}{13\!\cdots\!11}a^{12}+\frac{26\!\cdots\!85}{26\!\cdots\!22}a^{11}+\frac{11\!\cdots\!35}{26\!\cdots\!22}a^{10}+\frac{25\!\cdots\!45}{15\!\cdots\!66}a^{9}+\frac{56\!\cdots\!47}{52\!\cdots\!44}a^{8}-\frac{73\!\cdots\!63}{52\!\cdots\!44}a^{7}+\frac{35\!\cdots\!36}{13\!\cdots\!11}a^{6}+\frac{73\!\cdots\!25}{26\!\cdots\!22}a^{5}+\frac{52\!\cdots\!33}{26\!\cdots\!22}a^{4}+\frac{22\!\cdots\!26}{13\!\cdots\!11}a^{3}-\frac{13\!\cdots\!75}{13\!\cdots\!11}a^{2}-\frac{35\!\cdots\!07}{13\!\cdots\!11}a+\frac{14\!\cdots\!12}{13\!\cdots\!11}$, $\frac{88\!\cdots\!91}{52\!\cdots\!44}a^{20}-\frac{27\!\cdots\!89}{52\!\cdots\!44}a^{19}+\frac{10\!\cdots\!33}{52\!\cdots\!44}a^{18}-\frac{34\!\cdots\!73}{52\!\cdots\!44}a^{17}-\frac{36\!\cdots\!56}{13\!\cdots\!11}a^{16}-\frac{13\!\cdots\!22}{13\!\cdots\!11}a^{15}-\frac{62\!\cdots\!01}{13\!\cdots\!11}a^{14}+\frac{79\!\cdots\!72}{13\!\cdots\!11}a^{13}-\frac{96\!\cdots\!13}{26\!\cdots\!22}a^{12}+\frac{12\!\cdots\!67}{26\!\cdots\!22}a^{11}+\frac{22\!\cdots\!38}{13\!\cdots\!11}a^{10}+\frac{47\!\cdots\!10}{77\!\cdots\!83}a^{9}+\frac{36\!\cdots\!31}{52\!\cdots\!44}a^{8}-\frac{37\!\cdots\!27}{52\!\cdots\!44}a^{7}+\frac{36\!\cdots\!92}{13\!\cdots\!11}a^{6}+\frac{18\!\cdots\!00}{13\!\cdots\!11}a^{5}+\frac{87\!\cdots\!61}{26\!\cdots\!22}a^{4}+\frac{92\!\cdots\!94}{13\!\cdots\!11}a^{3}+\frac{10\!\cdots\!62}{13\!\cdots\!11}a^{2}-\frac{17\!\cdots\!33}{13\!\cdots\!11}a-\frac{20\!\cdots\!08}{13\!\cdots\!11}$, $\frac{21\!\cdots\!45}{21\!\cdots\!76}a^{20}-\frac{58\!\cdots\!51}{21\!\cdots\!76}a^{19}+\frac{57\!\cdots\!33}{21\!\cdots\!76}a^{18}-\frac{10\!\cdots\!39}{21\!\cdots\!76}a^{17}-\frac{80\!\cdots\!03}{52\!\cdots\!44}a^{16}+\frac{26\!\cdots\!31}{52\!\cdots\!44}a^{15}-\frac{17\!\cdots\!05}{52\!\cdots\!44}a^{14}+\frac{77\!\cdots\!93}{10\!\cdots\!88}a^{13}-\frac{19\!\cdots\!61}{52\!\cdots\!44}a^{12}+\frac{14\!\cdots\!83}{52\!\cdots\!44}a^{11}-\frac{18\!\cdots\!97}{52\!\cdots\!44}a^{10}+\frac{22\!\cdots\!37}{31\!\cdots\!32}a^{9}+\frac{13\!\cdots\!35}{21\!\cdots\!76}a^{8}-\frac{16\!\cdots\!93}{21\!\cdots\!76}a^{7}+\frac{81\!\cdots\!31}{52\!\cdots\!44}a^{6}-\frac{25\!\cdots\!45}{52\!\cdots\!44}a^{5}-\frac{55\!\cdots\!69}{26\!\cdots\!22}a^{4}+\frac{45\!\cdots\!96}{13\!\cdots\!11}a^{3}-\frac{28\!\cdots\!47}{13\!\cdots\!11}a^{2}-\frac{20\!\cdots\!75}{52\!\cdots\!44}a+\frac{34\!\cdots\!17}{52\!\cdots\!44}$
| 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: | \( 13512231302100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 13512231302100 \cdot 1}{2\cdot\sqrt{231636832837360510908018449295052404602048}}\cr\approx \mathstrut & 0.694644040655394 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900)
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^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900, 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^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900);
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^21 - 21*x^17 - 280*x^15 - 138*x^14 + 98*x^13 + 1792*x^12 + 2352*x^11 + 1008*x^10 + 13279*x^9 - 44576*x^8 + 4321*x^7 + 142296*x^6 - 106484*x^5 + 32928*x^4 + 355376*x^3 - 154252*x^2 - 254912*x + 83900);
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
$C_7^3:(C_3\times S_3)$ (as 21T40):
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 6174 |
| The 60 conjugacy class representatives for $C_7^3:(C_3\times S_3)$ |
| Character table for $C_7^3:(C_3\times S_3)$ |
Intermediate fields
| 3.1.107.1 |
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 21 siblings: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 21.7.8064534907239574094801006114055662600192.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $21$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $21$ | $21$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(7\)
| 7.7.10.3 | $x^{7} + 14 x^{4} + 7$ | $7$ | $1$ | $10$ | $C_7:C_3$ | $[5/3]_{3}$ |
| 7.14.20.2 | $x^{14} - 84 x^{12} - 196 x^{11} - 882 x^{10} - 588 x^{9} + 2254 x^{8} + 14 x^{7} - 588 x^{5} - 1372 x^{4} + 49$ | $7$ | $2$ | $20$ | 14T14 | $[5/3, 5/3]_{3}^{2}$ | |
|
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.3.0.1 | $x^{3} + 5 x + 105$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 107.3.0.1 | $x^{3} + 5 x + 105$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 107.6.3.2 | $x^{6} + 331 x^{4} + 210 x^{3} + 34372 x^{2} - 66360 x + 1124253$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 107.6.3.2 | $x^{6} + 331 x^{4} + 210 x^{3} + 34372 x^{2} - 66360 x + 1124253$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |