Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625)
gp: K = bnfinit(y^22 - 4*y^20 - 185*y^18 + 1908*y^16 - 5778*y^14 + 273*y^12 + 25663*y^10 - 37456*y^8 + 9890*y^6 + 8866*y^4 - 2350*y^2 - 625, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625)
\( x^{22} - 4 x^{20} - 185 x^{18} + 1908 x^{16} - 5778 x^{14} + 273 x^{12} + 25663 x^{10} - 37456 x^{8} + \cdots - 625 \)
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: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4129233136056857981979443884256982828952059904\)
\(\medspace = 2^{22}\cdot 74843^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.43\) | 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: |
\(2\), \(74843\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{17}-\frac{2}{5}a^{13}+\frac{2}{5}a^{11}-\frac{2}{5}a^{9}-\frac{2}{5}a^{7}-\frac{1}{5}a^{5}+\frac{1}{5}a$, $\frac{1}{11\!\cdots\!75}a^{20}+\frac{50\!\cdots\!21}{11\!\cdots\!75}a^{18}-\frac{12\!\cdots\!28}{24\!\cdots\!75}a^{16}+\frac{55\!\cdots\!62}{12\!\cdots\!75}a^{14}+\frac{84\!\cdots\!83}{12\!\cdots\!75}a^{12}-\frac{17\!\cdots\!84}{36\!\cdots\!25}a^{10}+\frac{53\!\cdots\!63}{11\!\cdots\!75}a^{8}-\frac{15\!\cdots\!06}{11\!\cdots\!75}a^{6}+\frac{19\!\cdots\!68}{22\!\cdots\!75}a^{4}-\frac{27\!\cdots\!01}{12\!\cdots\!75}a^{2}+\frac{18\!\cdots\!08}{44\!\cdots\!55}$, $\frac{1}{55\!\cdots\!75}a^{21}+\frac{50\!\cdots\!21}{55\!\cdots\!75}a^{19}-\frac{12\!\cdots\!28}{12\!\cdots\!75}a^{17}+\frac{17\!\cdots\!37}{61\!\cdots\!75}a^{15}-\frac{23\!\cdots\!67}{61\!\cdots\!75}a^{13}-\frac{90\!\cdots\!34}{18\!\cdots\!25}a^{11}+\frac{16\!\cdots\!38}{55\!\cdots\!75}a^{9}+\frac{21\!\cdots\!44}{55\!\cdots\!75}a^{7}+\frac{24\!\cdots\!43}{11\!\cdots\!75}a^{5}-\frac{27\!\cdots\!51}{61\!\cdots\!75}a^{3}+\frac{10\!\cdots\!18}{22\!\cdots\!75}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{99\!\cdots\!86}{61\!\cdots\!75}a^{21}-\frac{15\!\cdots\!44}{61\!\cdots\!75}a^{19}-\frac{38\!\cdots\!72}{12\!\cdots\!75}a^{17}+\frac{14\!\cdots\!38}{61\!\cdots\!75}a^{15}-\frac{15\!\cdots\!33}{61\!\cdots\!75}a^{13}-\frac{96\!\cdots\!72}{61\!\cdots\!75}a^{11}+\frac{17\!\cdots\!18}{61\!\cdots\!75}a^{9}+\frac{14\!\cdots\!59}{61\!\cdots\!75}a^{7}-\frac{61\!\cdots\!77}{12\!\cdots\!75}a^{5}+\frac{37\!\cdots\!01}{61\!\cdots\!75}a^{3}+\frac{10\!\cdots\!08}{24\!\cdots\!75}a$, $\frac{47\!\cdots\!74}{36\!\cdots\!25}a^{20}-\frac{23\!\cdots\!21}{36\!\cdots\!25}a^{18}-\frac{57\!\cdots\!66}{24\!\cdots\!75}a^{16}+\frac{32\!\cdots\!14}{12\!\cdots\!75}a^{14}-\frac{11\!\cdots\!99}{12\!\cdots\!75}a^{12}+\frac{44\!\cdots\!34}{12\!\cdots\!75}a^{10}+\frac{14\!\cdots\!37}{36\!\cdots\!25}a^{8}-\frac{24\!\cdots\!94}{36\!\cdots\!25}a^{6}+\frac{14\!\cdots\!07}{73\!\cdots\!25}a^{4}+\frac{19\!\cdots\!28}{12\!\cdots\!75}a^{2}-\frac{45\!\cdots\!33}{14\!\cdots\!85}$, $\frac{43\!\cdots\!66}{36\!\cdots\!25}a^{20}-\frac{13\!\cdots\!89}{36\!\cdots\!25}a^{18}-\frac{54\!\cdots\!94}{24\!\cdots\!75}a^{16}+\frac{25\!\cdots\!76}{12\!\cdots\!75}a^{14}-\frac{60\!\cdots\!16}{12\!\cdots\!75}a^{12}-\frac{48\!\cdots\!44}{12\!\cdots\!75}a^{10}+\frac{94\!\cdots\!83}{36\!\cdots\!25}a^{8}-\frac{79\!\cdots\!46}{36\!\cdots\!25}a^{6}-\frac{35\!\cdots\!37}{73\!\cdots\!25}a^{4}+\frac{10\!\cdots\!27}{12\!\cdots\!75}a^{2}+\frac{22\!\cdots\!68}{14\!\cdots\!85}$, $\frac{46\!\cdots\!33}{55\!\cdots\!75}a^{21}-\frac{19\!\cdots\!82}{55\!\cdots\!75}a^{19}-\frac{18\!\cdots\!99}{12\!\cdots\!75}a^{17}+\frac{10\!\cdots\!71}{61\!\cdots\!75}a^{15}-\frac{32\!\cdots\!86}{61\!\cdots\!75}a^{13}+\frac{32\!\cdots\!53}{18\!\cdots\!25}a^{11}+\frac{11\!\cdots\!54}{55\!\cdots\!75}a^{9}-\frac{20\!\cdots\!48}{55\!\cdots\!75}a^{7}+\frac{21\!\cdots\!69}{11\!\cdots\!75}a^{5}+\frac{53\!\cdots\!92}{61\!\cdots\!75}a^{3}-\frac{42\!\cdots\!56}{22\!\cdots\!75}a$, $\frac{67\!\cdots\!94}{18\!\cdots\!25}a^{21}-\frac{27\!\cdots\!51}{18\!\cdots\!25}a^{19}-\frac{82\!\cdots\!71}{12\!\cdots\!75}a^{17}+\frac{43\!\cdots\!84}{61\!\cdots\!75}a^{15}-\frac{13\!\cdots\!69}{61\!\cdots\!75}a^{13}+\frac{26\!\cdots\!04}{61\!\cdots\!75}a^{11}+\frac{16\!\cdots\!47}{18\!\cdots\!25}a^{9}-\frac{27\!\cdots\!64}{18\!\cdots\!25}a^{7}+\frac{23\!\cdots\!67}{36\!\cdots\!25}a^{5}+\frac{11\!\cdots\!68}{61\!\cdots\!75}a^{3}-\frac{90\!\cdots\!18}{73\!\cdots\!25}a$, $\frac{15\!\cdots\!57}{55\!\cdots\!75}a^{21}-\frac{68\!\cdots\!03}{55\!\cdots\!75}a^{19}-\frac{62\!\cdots\!96}{12\!\cdots\!75}a^{17}+\frac{33\!\cdots\!59}{61\!\cdots\!75}a^{15}-\frac{11\!\cdots\!69}{61\!\cdots\!75}a^{13}+\frac{12\!\cdots\!62}{18\!\cdots\!25}a^{11}+\frac{40\!\cdots\!41}{55\!\cdots\!75}a^{9}-\frac{73\!\cdots\!67}{55\!\cdots\!75}a^{7}+\frac{64\!\cdots\!01}{11\!\cdots\!75}a^{5}+\frac{13\!\cdots\!43}{61\!\cdots\!75}a^{3}-\frac{24\!\cdots\!89}{22\!\cdots\!75}a$, $\frac{13\!\cdots\!49}{11\!\cdots\!75}a^{20}-\frac{52\!\cdots\!46}{11\!\cdots\!75}a^{18}-\frac{54\!\cdots\!47}{24\!\cdots\!75}a^{16}+\frac{28\!\cdots\!63}{12\!\cdots\!75}a^{14}-\frac{83\!\cdots\!33}{12\!\cdots\!75}a^{12}+\frac{35\!\cdots\!09}{36\!\cdots\!25}a^{10}+\frac{33\!\cdots\!37}{11\!\cdots\!75}a^{8}-\frac{47\!\cdots\!94}{11\!\cdots\!75}a^{6}+\frac{29\!\cdots\!07}{22\!\cdots\!75}a^{4}+\frac{84\!\cdots\!26}{12\!\cdots\!75}a^{2}-\frac{10\!\cdots\!88}{44\!\cdots\!55}$, $\frac{58\!\cdots\!44}{11\!\cdots\!75}a^{20}-\frac{18\!\cdots\!76}{11\!\cdots\!75}a^{18}-\frac{24\!\cdots\!57}{24\!\cdots\!75}a^{16}+\frac{11\!\cdots\!53}{12\!\cdots\!75}a^{14}-\frac{28\!\cdots\!48}{12\!\cdots\!75}a^{12}-\frac{52\!\cdots\!96}{36\!\cdots\!25}a^{10}+\frac{12\!\cdots\!72}{11\!\cdots\!75}a^{8}-\frac{12\!\cdots\!64}{11\!\cdots\!75}a^{6}+\frac{66\!\cdots\!67}{22\!\cdots\!75}a^{4}+\frac{17\!\cdots\!56}{12\!\cdots\!75}a^{2}+\frac{12\!\cdots\!47}{44\!\cdots\!55}$, $\frac{15\!\cdots\!98}{11\!\cdots\!75}a^{20}-\frac{81\!\cdots\!42}{11\!\cdots\!75}a^{18}-\frac{60\!\cdots\!94}{24\!\cdots\!75}a^{16}+\frac{36\!\cdots\!01}{12\!\cdots\!75}a^{14}-\frac{13\!\cdots\!91}{12\!\cdots\!75}a^{12}+\frac{31\!\cdots\!43}{36\!\cdots\!25}a^{10}+\frac{44\!\cdots\!74}{11\!\cdots\!75}a^{8}-\frac{98\!\cdots\!38}{11\!\cdots\!75}a^{6}+\frac{11\!\cdots\!14}{22\!\cdots\!75}a^{4}+\frac{11\!\cdots\!27}{12\!\cdots\!75}a^{2}-\frac{85\!\cdots\!76}{44\!\cdots\!55}$, $\frac{49\!\cdots\!22}{12\!\cdots\!75}a^{21}-\frac{11\!\cdots\!63}{12\!\cdots\!75}a^{19}-\frac{18\!\cdots\!09}{24\!\cdots\!75}a^{17}+\frac{79\!\cdots\!26}{12\!\cdots\!75}a^{15}-\frac{15\!\cdots\!41}{12\!\cdots\!75}a^{13}-\frac{26\!\cdots\!94}{12\!\cdots\!75}a^{11}+\frac{91\!\cdots\!36}{12\!\cdots\!75}a^{9}-\frac{30\!\cdots\!82}{12\!\cdots\!75}a^{7}-\frac{70\!\cdots\!89}{24\!\cdots\!75}a^{5}-\frac{19\!\cdots\!98}{12\!\cdots\!75}a^{3}+\frac{39\!\cdots\!63}{49\!\cdots\!95}a$, $\frac{49\!\cdots\!09}{55\!\cdots\!75}a^{21}-\frac{99\!\cdots\!14}{11\!\cdots\!75}a^{20}-\frac{14\!\cdots\!61}{55\!\cdots\!75}a^{19}+\frac{28\!\cdots\!56}{11\!\cdots\!75}a^{18}-\frac{20\!\cdots\!27}{12\!\cdots\!75}a^{17}+\frac{41\!\cdots\!42}{24\!\cdots\!75}a^{16}+\frac{93\!\cdots\!58}{61\!\cdots\!75}a^{15}-\frac{18\!\cdots\!93}{12\!\cdots\!75}a^{14}-\frac{21\!\cdots\!53}{61\!\cdots\!75}a^{13}+\frac{42\!\cdots\!38}{12\!\cdots\!75}a^{12}-\frac{64\!\cdots\!31}{18\!\cdots\!25}a^{11}+\frac{13\!\cdots\!76}{36\!\cdots\!25}a^{10}+\frac{10\!\cdots\!17}{55\!\cdots\!75}a^{9}-\frac{20\!\cdots\!07}{11\!\cdots\!75}a^{8}-\frac{71\!\cdots\!04}{55\!\cdots\!75}a^{7}+\frac{13\!\cdots\!84}{11\!\cdots\!75}a^{6}-\frac{48\!\cdots\!13}{11\!\cdots\!75}a^{5}+\frac{11\!\cdots\!98}{22\!\cdots\!75}a^{4}+\frac{14\!\cdots\!41}{61\!\cdots\!75}a^{3}-\frac{28\!\cdots\!86}{12\!\cdots\!75}a^{2}+\frac{86\!\cdots\!22}{22\!\cdots\!75}a-\frac{24\!\cdots\!47}{44\!\cdots\!55}$, $\frac{12\!\cdots\!56}{18\!\cdots\!25}a^{21}+\frac{75\!\cdots\!99}{11\!\cdots\!75}a^{20}-\frac{34\!\cdots\!74}{18\!\cdots\!25}a^{19}-\frac{19\!\cdots\!71}{11\!\cdots\!75}a^{18}-\frac{16\!\cdots\!79}{12\!\cdots\!75}a^{17}-\frac{31\!\cdots\!97}{24\!\cdots\!75}a^{16}+\frac{70\!\cdots\!41}{61\!\cdots\!75}a^{15}+\frac{13\!\cdots\!88}{12\!\cdots\!75}a^{14}-\frac{15\!\cdots\!06}{61\!\cdots\!75}a^{13}-\frac{28\!\cdots\!58}{12\!\cdots\!75}a^{12}-\frac{18\!\cdots\!04}{61\!\cdots\!75}a^{11}-\frac{12\!\cdots\!66}{36\!\cdots\!25}a^{10}+\frac{25\!\cdots\!53}{18\!\cdots\!25}a^{9}+\frac{14\!\cdots\!87}{11\!\cdots\!75}a^{8}-\frac{15\!\cdots\!61}{18\!\cdots\!25}a^{7}-\frac{70\!\cdots\!94}{11\!\cdots\!75}a^{6}-\frac{91\!\cdots\!92}{36\!\cdots\!25}a^{5}-\frac{11\!\cdots\!93}{22\!\cdots\!75}a^{4}+\frac{80\!\cdots\!32}{61\!\cdots\!75}a^{3}+\frac{19\!\cdots\!01}{12\!\cdots\!75}a^{2}+\frac{72\!\cdots\!68}{73\!\cdots\!25}a+\frac{26\!\cdots\!67}{44\!\cdots\!55}$, $\frac{55\!\cdots\!24}{11\!\cdots\!75}a^{21}-\frac{15\!\cdots\!62}{11\!\cdots\!75}a^{20}-\frac{27\!\cdots\!71}{11\!\cdots\!75}a^{19}+\frac{43\!\cdots\!23}{11\!\cdots\!75}a^{18}-\frac{22\!\cdots\!12}{24\!\cdots\!75}a^{17}+\frac{66\!\cdots\!61}{24\!\cdots\!75}a^{16}+\frac{12\!\cdots\!38}{12\!\cdots\!75}a^{15}-\frac{29\!\cdots\!19}{12\!\cdots\!75}a^{14}-\frac{42\!\cdots\!58}{12\!\cdots\!75}a^{13}+\frac{66\!\cdots\!04}{12\!\cdots\!75}a^{12}+\frac{12\!\cdots\!09}{36\!\cdots\!25}a^{11}+\frac{19\!\cdots\!58}{36\!\cdots\!25}a^{10}+\frac{18\!\cdots\!62}{11\!\cdots\!75}a^{9}-\frac{31\!\cdots\!31}{11\!\cdots\!75}a^{8}-\frac{23\!\cdots\!69}{11\!\cdots\!75}a^{7}+\frac{25\!\cdots\!47}{11\!\cdots\!75}a^{6}-\frac{32\!\cdots\!08}{22\!\cdots\!75}a^{5}+\frac{11\!\cdots\!59}{22\!\cdots\!75}a^{4}+\frac{93\!\cdots\!76}{12\!\cdots\!75}a^{3}-\frac{79\!\cdots\!38}{12\!\cdots\!75}a^{2}+\frac{12\!\cdots\!99}{88\!\cdots\!31}a-\frac{52\!\cdots\!61}{44\!\cdots\!55}$, $\frac{52\!\cdots\!16}{18\!\cdots\!25}a^{21}+\frac{55\!\cdots\!83}{11\!\cdots\!75}a^{20}-\frac{22\!\cdots\!14}{18\!\cdots\!25}a^{19}-\frac{17\!\cdots\!32}{11\!\cdots\!75}a^{18}-\frac{64\!\cdots\!94}{12\!\cdots\!75}a^{17}-\frac{22\!\cdots\!24}{24\!\cdots\!75}a^{16}+\frac{34\!\cdots\!76}{61\!\cdots\!75}a^{15}+\frac{10\!\cdots\!71}{12\!\cdots\!75}a^{14}-\frac{10\!\cdots\!16}{61\!\cdots\!75}a^{13}-\frac{29\!\cdots\!86}{12\!\cdots\!75}a^{12}+\frac{11\!\cdots\!06}{61\!\cdots\!75}a^{11}-\frac{26\!\cdots\!72}{36\!\cdots\!25}a^{10}+\frac{13\!\cdots\!58}{18\!\cdots\!25}a^{9}+\frac{12\!\cdots\!79}{11\!\cdots\!75}a^{8}-\frac{20\!\cdots\!96}{18\!\cdots\!25}a^{7}-\frac{14\!\cdots\!48}{11\!\cdots\!75}a^{6}+\frac{12\!\cdots\!38}{36\!\cdots\!25}a^{5}+\frac{70\!\cdots\!69}{22\!\cdots\!75}a^{4}+\frac{77\!\cdots\!27}{61\!\cdots\!75}a^{3}+\frac{13\!\cdots\!42}{12\!\cdots\!75}a^{2}+\frac{38\!\cdots\!38}{73\!\cdots\!25}a+\frac{18\!\cdots\!44}{44\!\cdots\!55}$, $\frac{10\!\cdots\!57}{55\!\cdots\!75}a^{21}+\frac{33\!\cdots\!92}{11\!\cdots\!75}a^{20}-\frac{40\!\cdots\!28}{55\!\cdots\!75}a^{19}-\frac{10\!\cdots\!18}{11\!\cdots\!75}a^{18}-\frac{44\!\cdots\!21}{12\!\cdots\!75}a^{17}-\frac{13\!\cdots\!01}{24\!\cdots\!75}a^{16}+\frac{22\!\cdots\!09}{61\!\cdots\!75}a^{15}+\frac{64\!\cdots\!29}{12\!\cdots\!75}a^{14}-\frac{62\!\cdots\!94}{61\!\cdots\!75}a^{13}-\frac{16\!\cdots\!64}{12\!\cdots\!75}a^{12}-\frac{58\!\cdots\!63}{18\!\cdots\!25}a^{11}-\frac{34\!\cdots\!78}{36\!\cdots\!25}a^{10}+\frac{28\!\cdots\!66}{55\!\cdots\!75}a^{9}+\frac{75\!\cdots\!21}{11\!\cdots\!75}a^{8}-\frac{30\!\cdots\!92}{55\!\cdots\!75}a^{7}-\frac{64\!\cdots\!27}{11\!\cdots\!75}a^{6}-\frac{52\!\cdots\!49}{11\!\cdots\!75}a^{5}-\frac{25\!\cdots\!44}{22\!\cdots\!75}a^{4}+\frac{10\!\cdots\!93}{61\!\cdots\!75}a^{3}+\frac{19\!\cdots\!83}{12\!\cdots\!75}a^{2}+\frac{69\!\cdots\!81}{22\!\cdots\!75}a+\frac{12\!\cdots\!81}{44\!\cdots\!55}$, $\frac{17\!\cdots\!43}{18\!\cdots\!25}a^{21}+\frac{99\!\cdots\!68}{11\!\cdots\!75}a^{20}-\frac{49\!\cdots\!97}{18\!\cdots\!25}a^{19}-\frac{28\!\cdots\!72}{11\!\cdots\!75}a^{18}-\frac{21\!\cdots\!12}{12\!\cdots\!75}a^{17}-\frac{41\!\cdots\!79}{24\!\cdots\!75}a^{16}+\frac{97\!\cdots\!98}{61\!\cdots\!75}a^{15}+\frac{18\!\cdots\!91}{12\!\cdots\!75}a^{14}-\frac{22\!\cdots\!93}{61\!\cdots\!75}a^{13}-\frac{42\!\cdots\!56}{12\!\cdots\!75}a^{12}-\frac{22\!\cdots\!62}{61\!\cdots\!75}a^{11}-\frac{13\!\cdots\!12}{36\!\cdots\!25}a^{10}+\frac{35\!\cdots\!59}{18\!\cdots\!25}a^{9}+\frac{20\!\cdots\!09}{11\!\cdots\!75}a^{8}-\frac{24\!\cdots\!58}{18\!\cdots\!25}a^{7}-\frac{13\!\cdots\!33}{11\!\cdots\!75}a^{6}-\frac{14\!\cdots\!51}{36\!\cdots\!25}a^{5}-\frac{85\!\cdots\!26}{22\!\cdots\!75}a^{4}+\frac{15\!\cdots\!46}{61\!\cdots\!75}a^{3}+\frac{31\!\cdots\!07}{12\!\cdots\!75}a^{2}+\frac{36\!\cdots\!09}{73\!\cdots\!25}a+\frac{20\!\cdots\!94}{44\!\cdots\!55}$, $\frac{72\!\cdots\!23}{55\!\cdots\!75}a^{21}+\frac{14\!\cdots\!88}{11\!\cdots\!75}a^{20}-\frac{21\!\cdots\!92}{55\!\cdots\!75}a^{19}-\frac{42\!\cdots\!27}{11\!\cdots\!75}a^{18}-\frac{30\!\cdots\!44}{12\!\cdots\!75}a^{17}-\frac{61\!\cdots\!14}{24\!\cdots\!75}a^{16}+\frac{13\!\cdots\!51}{61\!\cdots\!75}a^{15}+\frac{27\!\cdots\!31}{12\!\cdots\!75}a^{14}-\frac{32\!\cdots\!16}{61\!\cdots\!75}a^{13}-\frac{63\!\cdots\!96}{12\!\cdots\!75}a^{12}-\frac{87\!\cdots\!32}{18\!\cdots\!25}a^{11}-\frac{19\!\cdots\!67}{36\!\cdots\!25}a^{10}+\frac{15\!\cdots\!99}{55\!\cdots\!75}a^{9}+\frac{30\!\cdots\!94}{11\!\cdots\!75}a^{8}-\frac{11\!\cdots\!63}{55\!\cdots\!75}a^{7}-\frac{21\!\cdots\!03}{11\!\cdots\!75}a^{6}-\frac{60\!\cdots\!86}{11\!\cdots\!75}a^{5}-\frac{13\!\cdots\!16}{22\!\cdots\!75}a^{4}+\frac{33\!\cdots\!27}{61\!\cdots\!75}a^{3}+\frac{56\!\cdots\!37}{12\!\cdots\!75}a^{2}+\frac{22\!\cdots\!49}{22\!\cdots\!75}a+\frac{39\!\cdots\!29}{44\!\cdots\!55}$
| 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: | \( 1955163891580000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 1955163891580000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 0.388470355618626 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625)
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 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625, 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 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625);
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 - 4*x^20 - 185*x^18 + 1908*x^16 - 5778*x^14 + 273*x^12 + 25663*x^10 - 37456*x^8 + 9890*x^6 + 8866*x^4 - 2350*x^2 - 625);
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
$C_2^{10}.\PSL(2,11)$ (as 22T39):
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 non-solvable group of order 675840 |
| The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$ |
| Character table for $C_2^{10}.\PSL(2,11)$ |
Intermediate fields
| 11.11.31376518243389673201.1 |
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 |
| Minimal sibling: | 22.6.4129233136056857981979443884256982828952059904.6 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(74843\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |