Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369)
gp: K = bnfinit(y^22 + 12*y^20 - 361*y^18 - 3364*y^16 + 27305*y^14 + 69776*y^12 - 559114*y^10 + 358013*y^8 + 751769*y^6 + 204573*y^4 - 6542*y^2 - 1369, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369)
\( x^{22} + 12 x^{20} - 361 x^{18} - 3364 x^{16} + 27305 x^{14} + 69776 x^{12} - 559114 x^{10} + \cdots - 1369 \)
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: |
\(4129233136056857981979443884256982828952059904\)
\(\medspace = 2^{22}\cdot 74843^{8}\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{22\!\cdots\!75}a^{20}-\frac{11\!\cdots\!66}{22\!\cdots\!75}a^{18}-\frac{17\!\cdots\!96}{73\!\cdots\!25}a^{16}+\frac{28\!\cdots\!16}{88\!\cdots\!63}a^{14}+\frac{20\!\cdots\!26}{44\!\cdots\!15}a^{12}+\frac{15\!\cdots\!87}{73\!\cdots\!25}a^{10}+\frac{78\!\cdots\!03}{22\!\cdots\!75}a^{8}+\frac{56\!\cdots\!79}{22\!\cdots\!75}a^{6}-\frac{46\!\cdots\!56}{73\!\cdots\!25}a^{4}+\frac{64\!\cdots\!09}{73\!\cdots\!25}a^{2}+\frac{17\!\cdots\!52}{22\!\cdots\!75}$, $\frac{1}{81\!\cdots\!75}a^{21}+\frac{20\!\cdots\!09}{81\!\cdots\!75}a^{19}-\frac{13\!\cdots\!46}{27\!\cdots\!25}a^{17}-\frac{10\!\cdots\!40}{32\!\cdots\!31}a^{15}-\frac{63\!\cdots\!99}{16\!\cdots\!55}a^{13}+\frac{44\!\cdots\!37}{27\!\cdots\!25}a^{11}-\frac{19\!\cdots\!72}{81\!\cdots\!75}a^{9}-\frac{32\!\cdots\!46}{81\!\cdots\!75}a^{7}+\frac{80\!\cdots\!19}{27\!\cdots\!25}a^{5}+\frac{29\!\cdots\!32}{73\!\cdots\!25}a^{3}+\frac{11\!\cdots\!27}{81\!\cdots\!75}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{10\!\cdots\!34}{27\!\cdots\!25}a^{21}+\frac{12\!\cdots\!81}{27\!\cdots\!25}a^{19}-\frac{37\!\cdots\!42}{27\!\cdots\!25}a^{17}-\frac{14\!\cdots\!40}{10\!\cdots\!77}a^{15}+\frac{54\!\cdots\!04}{54\!\cdots\!85}a^{13}+\frac{79\!\cdots\!24}{27\!\cdots\!25}a^{11}-\frac{56\!\cdots\!48}{27\!\cdots\!25}a^{9}+\frac{21\!\cdots\!36}{27\!\cdots\!25}a^{7}+\frac{91\!\cdots\!63}{27\!\cdots\!25}a^{5}+\frac{98\!\cdots\!39}{73\!\cdots\!25}a^{3}+\frac{15\!\cdots\!18}{27\!\cdots\!25}a$, $\frac{97\!\cdots\!83}{73\!\cdots\!25}a^{20}+\frac{11\!\cdots\!72}{73\!\cdots\!25}a^{18}-\frac{35\!\cdots\!29}{73\!\cdots\!25}a^{16}-\frac{12\!\cdots\!13}{29\!\cdots\!21}a^{14}+\frac{55\!\cdots\!28}{14\!\cdots\!05}a^{12}+\frac{56\!\cdots\!13}{73\!\cdots\!25}a^{10}-\frac{56\!\cdots\!01}{73\!\cdots\!25}a^{8}+\frac{58\!\cdots\!07}{73\!\cdots\!25}a^{6}+\frac{48\!\cdots\!31}{73\!\cdots\!25}a^{4}+\frac{99\!\cdots\!16}{73\!\cdots\!25}a^{2}-\frac{25\!\cdots\!34}{73\!\cdots\!25}$, $\frac{40\!\cdots\!16}{73\!\cdots\!25}a^{20}+\frac{51\!\cdots\!19}{73\!\cdots\!25}a^{18}-\frac{14\!\cdots\!08}{73\!\cdots\!25}a^{16}-\frac{58\!\cdots\!21}{29\!\cdots\!21}a^{14}+\frac{20\!\cdots\!31}{14\!\cdots\!05}a^{12}+\frac{35\!\cdots\!76}{73\!\cdots\!25}a^{10}-\frac{21\!\cdots\!52}{73\!\cdots\!25}a^{8}+\frac{78\!\cdots\!64}{73\!\cdots\!25}a^{6}+\frac{43\!\cdots\!12}{73\!\cdots\!25}a^{4}+\frac{20\!\cdots\!82}{73\!\cdots\!25}a^{2}-\frac{62\!\cdots\!93}{73\!\cdots\!25}$, $\frac{87\!\cdots\!04}{81\!\cdots\!75}a^{21}+\frac{10\!\cdots\!61}{81\!\cdots\!75}a^{19}-\frac{10\!\cdots\!09}{27\!\cdots\!25}a^{17}-\frac{11\!\cdots\!22}{32\!\cdots\!31}a^{15}+\frac{48\!\cdots\!84}{16\!\cdots\!55}a^{13}+\frac{19\!\cdots\!48}{27\!\cdots\!25}a^{11}-\frac{49\!\cdots\!38}{81\!\cdots\!75}a^{9}+\frac{38\!\cdots\!41}{81\!\cdots\!75}a^{7}+\frac{19\!\cdots\!26}{27\!\cdots\!25}a^{5}+\frac{10\!\cdots\!78}{73\!\cdots\!25}a^{3}-\frac{12\!\cdots\!17}{81\!\cdots\!75}a$, $\frac{10\!\cdots\!07}{81\!\cdots\!75}a^{21}+\frac{12\!\cdots\!13}{81\!\cdots\!75}a^{19}-\frac{13\!\cdots\!97}{27\!\cdots\!25}a^{17}-\frac{14\!\cdots\!69}{32\!\cdots\!31}a^{15}+\frac{62\!\cdots\!87}{16\!\cdots\!55}a^{13}+\frac{21\!\cdots\!84}{27\!\cdots\!25}a^{11}-\frac{63\!\cdots\!29}{81\!\cdots\!75}a^{9}+\frac{65\!\cdots\!03}{81\!\cdots\!75}a^{7}+\frac{17\!\cdots\!08}{27\!\cdots\!25}a^{5}+\frac{40\!\cdots\!99}{73\!\cdots\!25}a^{3}+\frac{12\!\cdots\!14}{81\!\cdots\!75}a$, $\frac{12\!\cdots\!84}{10\!\cdots\!77}a^{21}+\frac{14\!\cdots\!54}{10\!\cdots\!77}a^{19}-\frac{45\!\cdots\!20}{10\!\cdots\!77}a^{17}-\frac{40\!\cdots\!09}{10\!\cdots\!77}a^{15}+\frac{35\!\cdots\!79}{10\!\cdots\!77}a^{13}+\frac{76\!\cdots\!56}{10\!\cdots\!77}a^{11}-\frac{71\!\cdots\!94}{10\!\cdots\!77}a^{9}+\frac{65\!\cdots\!90}{10\!\cdots\!77}a^{7}+\frac{72\!\cdots\!87}{10\!\cdots\!77}a^{5}+\frac{22\!\cdots\!46}{29\!\cdots\!21}a^{3}-\frac{18\!\cdots\!20}{10\!\cdots\!77}a$, $\frac{18\!\cdots\!83}{27\!\cdots\!25}a^{21}+\frac{21\!\cdots\!22}{27\!\cdots\!25}a^{19}-\frac{67\!\cdots\!04}{27\!\cdots\!25}a^{17}-\frac{23\!\cdots\!51}{10\!\cdots\!77}a^{15}+\frac{10\!\cdots\!68}{54\!\cdots\!85}a^{13}+\frac{10\!\cdots\!38}{27\!\cdots\!25}a^{11}-\frac{10\!\cdots\!01}{27\!\cdots\!25}a^{9}+\frac{11\!\cdots\!57}{27\!\cdots\!25}a^{7}+\frac{85\!\cdots\!56}{27\!\cdots\!25}a^{5}+\frac{36\!\cdots\!93}{73\!\cdots\!25}a^{3}-\frac{64\!\cdots\!09}{27\!\cdots\!25}a$, $\frac{84\!\cdots\!81}{22\!\cdots\!75}a^{20}+\frac{10\!\cdots\!79}{22\!\cdots\!75}a^{18}-\frac{10\!\cdots\!26}{73\!\cdots\!25}a^{16}-\frac{11\!\cdots\!24}{88\!\cdots\!63}a^{14}+\frac{46\!\cdots\!41}{44\!\cdots\!15}a^{12}+\frac{19\!\cdots\!72}{73\!\cdots\!25}a^{10}-\frac{47\!\cdots\!57}{22\!\cdots\!75}a^{8}+\frac{33\!\cdots\!74}{22\!\cdots\!75}a^{6}+\frac{19\!\cdots\!39}{73\!\cdots\!25}a^{4}+\frac{53\!\cdots\!79}{73\!\cdots\!25}a^{2}+\frac{54\!\cdots\!37}{22\!\cdots\!75}$, $\frac{26\!\cdots\!12}{81\!\cdots\!75}a^{21}+\frac{31\!\cdots\!58}{81\!\cdots\!75}a^{19}-\frac{32\!\cdots\!27}{27\!\cdots\!25}a^{17}-\frac{35\!\cdots\!70}{32\!\cdots\!31}a^{15}+\frac{14\!\cdots\!82}{16\!\cdots\!55}a^{13}+\frac{60\!\cdots\!69}{27\!\cdots\!25}a^{11}-\frac{14\!\cdots\!89}{81\!\cdots\!75}a^{9}+\frac{10\!\cdots\!73}{81\!\cdots\!75}a^{7}+\frac{64\!\cdots\!53}{27\!\cdots\!25}a^{5}+\frac{36\!\cdots\!84}{73\!\cdots\!25}a^{3}-\frac{46\!\cdots\!76}{81\!\cdots\!75}a$, $\frac{88\!\cdots\!42}{27\!\cdots\!25}a^{21}+\frac{10\!\cdots\!78}{27\!\cdots\!25}a^{19}-\frac{31\!\cdots\!46}{27\!\cdots\!25}a^{17}-\frac{12\!\cdots\!63}{10\!\cdots\!77}a^{15}+\frac{47\!\cdots\!57}{54\!\cdots\!85}a^{13}+\frac{66\!\cdots\!12}{27\!\cdots\!25}a^{11}-\frac{48\!\cdots\!24}{27\!\cdots\!25}a^{9}+\frac{21\!\cdots\!68}{27\!\cdots\!25}a^{7}+\frac{75\!\cdots\!19}{27\!\cdots\!25}a^{5}+\frac{78\!\cdots\!57}{73\!\cdots\!25}a^{3}+\frac{18\!\cdots\!59}{27\!\cdots\!25}a$, $\frac{13\!\cdots\!38}{32\!\cdots\!75}a^{21}-\frac{18\!\cdots\!91}{14\!\cdots\!05}a^{20}+\frac{16\!\cdots\!92}{32\!\cdots\!75}a^{19}-\frac{22\!\cdots\!59}{14\!\cdots\!05}a^{18}-\frac{50\!\cdots\!19}{32\!\cdots\!75}a^{17}+\frac{66\!\cdots\!93}{14\!\cdots\!05}a^{16}-\frac{18\!\cdots\!91}{13\!\cdots\!19}a^{15}+\frac{12\!\cdots\!38}{29\!\cdots\!21}a^{14}+\frac{75\!\cdots\!03}{65\!\cdots\!95}a^{13}-\frac{99\!\cdots\!13}{29\!\cdots\!21}a^{12}+\frac{10\!\cdots\!43}{32\!\cdots\!75}a^{11}-\frac{13\!\cdots\!41}{14\!\cdots\!05}a^{10}-\frac{77\!\cdots\!11}{32\!\cdots\!75}a^{9}+\frac{10\!\cdots\!82}{14\!\cdots\!05}a^{8}+\frac{43\!\cdots\!27}{32\!\cdots\!75}a^{7}-\frac{57\!\cdots\!39}{14\!\cdots\!05}a^{6}+\frac{10\!\cdots\!41}{32\!\cdots\!75}a^{5}-\frac{14\!\cdots\!82}{14\!\cdots\!05}a^{4}+\frac{10\!\cdots\!48}{88\!\cdots\!75}a^{3}-\frac{49\!\cdots\!62}{14\!\cdots\!05}a^{2}+\frac{22\!\cdots\!76}{32\!\cdots\!75}a-\frac{29\!\cdots\!07}{14\!\cdots\!05}$, $\frac{12\!\cdots\!61}{81\!\cdots\!75}a^{21}-\frac{13\!\cdots\!09}{22\!\cdots\!75}a^{20}+\frac{15\!\cdots\!24}{81\!\cdots\!75}a^{19}-\frac{17\!\cdots\!06}{22\!\cdots\!75}a^{18}-\frac{14\!\cdots\!31}{27\!\cdots\!25}a^{17}+\frac{15\!\cdots\!39}{73\!\cdots\!25}a^{16}-\frac{16\!\cdots\!30}{32\!\cdots\!31}a^{15}+\frac{19\!\cdots\!38}{88\!\cdots\!63}a^{14}+\frac{63\!\cdots\!06}{16\!\cdots\!55}a^{13}-\frac{61\!\cdots\!74}{44\!\cdots\!15}a^{12}+\frac{31\!\cdots\!82}{27\!\cdots\!25}a^{11}-\frac{39\!\cdots\!83}{73\!\cdots\!25}a^{10}-\frac{64\!\cdots\!17}{81\!\cdots\!75}a^{9}+\frac{59\!\cdots\!23}{22\!\cdots\!75}a^{8}+\frac{29\!\cdots\!94}{81\!\cdots\!75}a^{7}-\frac{25\!\cdots\!61}{22\!\cdots\!75}a^{6}+\frac{27\!\cdots\!34}{27\!\cdots\!25}a^{5}-\frac{19\!\cdots\!71}{73\!\cdots\!25}a^{4}+\frac{26\!\cdots\!27}{73\!\cdots\!25}a^{3}-\frac{78\!\cdots\!56}{73\!\cdots\!25}a^{2}+\frac{18\!\cdots\!22}{81\!\cdots\!75}a-\frac{14\!\cdots\!68}{22\!\cdots\!75}$, $\frac{27\!\cdots\!07}{27\!\cdots\!25}a^{21}-\frac{43\!\cdots\!41}{22\!\cdots\!75}a^{20}+\frac{43\!\cdots\!63}{27\!\cdots\!25}a^{19}-\frac{67\!\cdots\!19}{22\!\cdots\!75}a^{18}-\frac{84\!\cdots\!91}{27\!\cdots\!25}a^{17}+\frac{43\!\cdots\!86}{73\!\cdots\!25}a^{16}-\frac{49\!\cdots\!73}{10\!\cdots\!77}a^{15}+\frac{76\!\cdots\!67}{88\!\cdots\!63}a^{14}+\frac{61\!\cdots\!62}{54\!\cdots\!85}a^{13}-\frac{95\!\cdots\!36}{44\!\cdots\!15}a^{12}+\frac{30\!\cdots\!27}{27\!\cdots\!25}a^{11}-\frac{15\!\cdots\!17}{73\!\cdots\!25}a^{10}-\frac{44\!\cdots\!29}{27\!\cdots\!25}a^{9}+\frac{68\!\cdots\!02}{22\!\cdots\!75}a^{8}-\frac{61\!\cdots\!97}{27\!\cdots\!25}a^{7}+\frac{95\!\cdots\!11}{22\!\cdots\!75}a^{6}-\frac{15\!\cdots\!51}{27\!\cdots\!25}a^{5}+\frac{79\!\cdots\!21}{73\!\cdots\!25}a^{4}+\frac{15\!\cdots\!47}{73\!\cdots\!25}a^{3}-\frac{27\!\cdots\!19}{73\!\cdots\!25}a^{2}+\frac{10\!\cdots\!39}{27\!\cdots\!25}a-\frac{16\!\cdots\!32}{22\!\cdots\!75}$, $\frac{61\!\cdots\!03}{81\!\cdots\!75}a^{21}-\frac{47\!\cdots\!74}{22\!\cdots\!75}a^{20}+\frac{87\!\cdots\!52}{81\!\cdots\!75}a^{19}-\frac{77\!\cdots\!66}{22\!\cdots\!75}a^{18}-\frac{67\!\cdots\!88}{27\!\cdots\!25}a^{17}+\frac{44\!\cdots\!79}{73\!\cdots\!25}a^{16}-\frac{99\!\cdots\!08}{32\!\cdots\!31}a^{15}+\frac{87\!\cdots\!63}{88\!\cdots\!63}a^{14}+\frac{22\!\cdots\!58}{16\!\cdots\!55}a^{13}-\frac{58\!\cdots\!99}{44\!\cdots\!15}a^{12}+\frac{20\!\cdots\!36}{27\!\cdots\!25}a^{11}-\frac{14\!\cdots\!63}{73\!\cdots\!25}a^{10}-\frac{20\!\cdots\!91}{81\!\cdots\!75}a^{9}+\frac{36\!\cdots\!03}{22\!\cdots\!75}a^{8}-\frac{54\!\cdots\!88}{81\!\cdots\!75}a^{7}-\frac{85\!\cdots\!96}{22\!\cdots\!75}a^{6}+\frac{41\!\cdots\!57}{27\!\cdots\!25}a^{5}-\frac{43\!\cdots\!81}{73\!\cdots\!25}a^{4}+\frac{49\!\cdots\!71}{73\!\cdots\!25}a^{3}-\frac{13\!\cdots\!41}{73\!\cdots\!25}a^{2}+\frac{34\!\cdots\!56}{81\!\cdots\!75}a-\frac{24\!\cdots\!48}{22\!\cdots\!75}$, $\frac{54\!\cdots\!59}{81\!\cdots\!75}a^{21}+\frac{84\!\cdots\!53}{44\!\cdots\!15}a^{20}+\frac{65\!\cdots\!31}{81\!\cdots\!75}a^{19}+\frac{10\!\cdots\!47}{44\!\cdots\!15}a^{18}-\frac{64\!\cdots\!64}{27\!\cdots\!25}a^{17}-\frac{10\!\cdots\!88}{14\!\cdots\!05}a^{16}-\frac{73\!\cdots\!08}{32\!\cdots\!31}a^{15}-\frac{57\!\cdots\!12}{88\!\cdots\!63}a^{14}+\frac{29\!\cdots\!79}{16\!\cdots\!55}a^{13}+\frac{45\!\cdots\!65}{88\!\cdots\!63}a^{12}+\frac{12\!\cdots\!33}{27\!\cdots\!25}a^{11}+\frac{20\!\cdots\!26}{14\!\cdots\!05}a^{10}-\frac{29\!\cdots\!73}{81\!\cdots\!75}a^{9}-\frac{46\!\cdots\!81}{44\!\cdots\!15}a^{8}+\frac{16\!\cdots\!36}{81\!\cdots\!75}a^{7}+\frac{26\!\cdots\!77}{44\!\cdots\!15}a^{6}+\frac{14\!\cdots\!21}{27\!\cdots\!25}a^{5}+\frac{22\!\cdots\!67}{14\!\cdots\!05}a^{4}+\frac{13\!\cdots\!63}{73\!\cdots\!25}a^{3}+\frac{76\!\cdots\!27}{14\!\cdots\!05}a^{2}+\frac{87\!\cdots\!43}{81\!\cdots\!75}a+\frac{13\!\cdots\!51}{44\!\cdots\!15}$
| 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: | \( 3536795070310000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 3536795070310000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 1.73390127437880 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369)
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 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369, 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 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369);
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 + 12*x^20 - 361*x^18 - 3364*x^16 + 27305*x^14 + 69776*x^12 - 559114*x^10 + 358013*x^8 + 751769*x^6 + 204573*x^4 - 6542*x^2 - 1369);
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: | 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(74843\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |