Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 12*x^20 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129)
gp: K = bnfinit(y^22 + 12*y^20 - 192*y^18 - 3910*y^16 - 18357*y^14 + 37519*y^12 + 608582*y^10 + 2113799*y^8 + 3221580*y^6 + 1933376*y^4 - 120888*y^2 - 388129, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 12*x^20 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 12*x^20 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129)
\( x^{22} + 12 x^{20} - 192 x^{18} - 3910 x^{16} - 18357 x^{14} + 37519 x^{12} + 608582 x^{10} + \cdots - 388129 \)
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: | $[6, 8]$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{14}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{32\!\cdots\!39}a^{20}+\frac{11\!\cdots\!03}{10\!\cdots\!13}a^{18}-\frac{93\!\cdots\!99}{10\!\cdots\!13}a^{16}-\frac{44\!\cdots\!68}{10\!\cdots\!13}a^{14}+\frac{15\!\cdots\!16}{32\!\cdots\!39}a^{12}+\frac{54\!\cdots\!72}{32\!\cdots\!39}a^{10}+\frac{50\!\cdots\!82}{10\!\cdots\!13}a^{8}-\frac{25\!\cdots\!85}{10\!\cdots\!13}a^{6}+\frac{53\!\cdots\!04}{32\!\cdots\!39}a^{4}-\frac{70\!\cdots\!78}{32\!\cdots\!39}a^{2}-\frac{68\!\cdots\!45}{32\!\cdots\!39}$, $\frac{1}{20\!\cdots\!97}a^{21}-\frac{67\!\cdots\!03}{67\!\cdots\!99}a^{19}-\frac{43\!\cdots\!19}{67\!\cdots\!99}a^{17}+\frac{51\!\cdots\!56}{67\!\cdots\!99}a^{15}+\frac{74\!\cdots\!69}{20\!\cdots\!97}a^{13}+\frac{34\!\cdots\!67}{20\!\cdots\!97}a^{11}-\frac{37\!\cdots\!15}{76\!\cdots\!91}a^{9}-\frac{26\!\cdots\!97}{67\!\cdots\!99}a^{7}+\frac{39\!\cdots\!84}{20\!\cdots\!97}a^{5}-\frac{58\!\cdots\!98}{20\!\cdots\!97}a^{3}-\frac{80\!\cdots\!39}{20\!\cdots\!97}a$
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: | $13$ | 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: |
$\frac{49\!\cdots\!66}{32\!\cdots\!39}a^{20}+\frac{52\!\cdots\!36}{32\!\cdots\!39}a^{18}-\frac{10\!\cdots\!87}{32\!\cdots\!39}a^{16}-\frac{60\!\cdots\!58}{10\!\cdots\!13}a^{14}-\frac{66\!\cdots\!05}{32\!\cdots\!39}a^{12}+\frac{92\!\cdots\!11}{10\!\cdots\!13}a^{10}+\frac{26\!\cdots\!38}{32\!\cdots\!39}a^{8}+\frac{22\!\cdots\!79}{10\!\cdots\!13}a^{6}+\frac{65\!\cdots\!20}{32\!\cdots\!39}a^{4}+\frac{67\!\cdots\!36}{32\!\cdots\!39}a^{2}-\frac{14\!\cdots\!23}{32\!\cdots\!39}$, $\frac{57\!\cdots\!28}{32\!\cdots\!39}a^{20}+\frac{65\!\cdots\!78}{32\!\cdots\!39}a^{18}-\frac{11\!\cdots\!65}{32\!\cdots\!39}a^{16}-\frac{21\!\cdots\!08}{32\!\cdots\!39}a^{14}-\frac{30\!\cdots\!30}{10\!\cdots\!13}a^{12}+\frac{28\!\cdots\!29}{32\!\cdots\!39}a^{10}+\frac{33\!\cdots\!41}{32\!\cdots\!39}a^{8}+\frac{99\!\cdots\!84}{32\!\cdots\!39}a^{6}+\frac{37\!\cdots\!51}{10\!\cdots\!13}a^{4}+\frac{23\!\cdots\!37}{32\!\cdots\!39}a^{2}-\frac{92\!\cdots\!24}{10\!\cdots\!13}$, $\frac{45\!\cdots\!80}{20\!\cdots\!97}a^{21}+\frac{15\!\cdots\!29}{67\!\cdots\!99}a^{19}-\frac{31\!\cdots\!50}{67\!\cdots\!99}a^{17}-\frac{54\!\cdots\!06}{67\!\cdots\!99}a^{15}-\frac{57\!\cdots\!72}{20\!\cdots\!97}a^{13}+\frac{26\!\cdots\!02}{20\!\cdots\!97}a^{11}+\frac{88\!\cdots\!93}{76\!\cdots\!91}a^{9}+\frac{19\!\cdots\!70}{67\!\cdots\!99}a^{7}+\frac{52\!\cdots\!51}{20\!\cdots\!97}a^{5}+\frac{24\!\cdots\!43}{20\!\cdots\!97}a^{3}-\frac{10\!\cdots\!00}{20\!\cdots\!97}a$, $\frac{11\!\cdots\!73}{67\!\cdots\!99}a^{21}+\frac{36\!\cdots\!22}{20\!\cdots\!97}a^{19}-\frac{72\!\cdots\!98}{20\!\cdots\!97}a^{17}-\frac{41\!\cdots\!61}{67\!\cdots\!99}a^{15}-\frac{44\!\cdots\!18}{20\!\cdots\!97}a^{13}+\frac{20\!\cdots\!49}{20\!\cdots\!97}a^{11}+\frac{20\!\cdots\!18}{22\!\cdots\!73}a^{9}+\frac{15\!\cdots\!43}{67\!\cdots\!99}a^{7}+\frac{40\!\cdots\!56}{20\!\cdots\!97}a^{5}+\frac{24\!\cdots\!85}{20\!\cdots\!97}a^{3}-\frac{28\!\cdots\!29}{67\!\cdots\!99}a$, $\frac{70\!\cdots\!67}{32\!\cdots\!39}a^{20}+\frac{24\!\cdots\!59}{10\!\cdots\!13}a^{18}-\frac{48\!\cdots\!22}{10\!\cdots\!13}a^{16}-\frac{83\!\cdots\!45}{10\!\cdots\!13}a^{14}-\frac{88\!\cdots\!88}{32\!\cdots\!39}a^{12}+\frac{41\!\cdots\!82}{32\!\cdots\!39}a^{10}+\frac{12\!\cdots\!61}{10\!\cdots\!13}a^{8}+\frac{29\!\cdots\!46}{10\!\cdots\!13}a^{6}+\frac{77\!\cdots\!24}{32\!\cdots\!39}a^{4}+\frac{26\!\cdots\!21}{32\!\cdots\!39}a^{2}-\frac{17\!\cdots\!89}{32\!\cdots\!39}$, $\frac{11\!\cdots\!43}{67\!\cdots\!99}a^{21}+\frac{35\!\cdots\!80}{20\!\cdots\!97}a^{19}-\frac{24\!\cdots\!44}{67\!\cdots\!99}a^{17}-\frac{12\!\cdots\!79}{20\!\cdots\!97}a^{15}-\frac{42\!\cdots\!54}{20\!\cdots\!97}a^{13}+\frac{19\!\cdots\!09}{20\!\cdots\!97}a^{11}+\frac{64\!\cdots\!56}{76\!\cdots\!91}a^{9}+\frac{43\!\cdots\!53}{20\!\cdots\!97}a^{7}+\frac{41\!\cdots\!26}{20\!\cdots\!97}a^{5}+\frac{47\!\cdots\!56}{20\!\cdots\!97}a^{3}-\frac{98\!\cdots\!96}{20\!\cdots\!97}a$, $\frac{26\!\cdots\!59}{20\!\cdots\!97}a^{21}+\frac{94\!\cdots\!66}{67\!\cdots\!99}a^{19}-\frac{53\!\cdots\!14}{20\!\cdots\!97}a^{17}-\frac{95\!\cdots\!93}{20\!\cdots\!97}a^{15}-\frac{36\!\cdots\!58}{20\!\cdots\!97}a^{13}+\frac{47\!\cdots\!47}{67\!\cdots\!99}a^{11}+\frac{15\!\cdots\!53}{22\!\cdots\!73}a^{9}+\frac{37\!\cdots\!30}{20\!\cdots\!97}a^{7}+\frac{37\!\cdots\!16}{20\!\cdots\!97}a^{5}+\frac{57\!\cdots\!82}{20\!\cdots\!97}a^{3}-\frac{22\!\cdots\!74}{67\!\cdots\!99}a$, $\frac{22\!\cdots\!50}{10\!\cdots\!13}a^{20}+\frac{70\!\cdots\!15}{32\!\cdots\!39}a^{18}-\frac{13\!\cdots\!24}{32\!\cdots\!39}a^{16}-\frac{24\!\cdots\!93}{32\!\cdots\!39}a^{14}-\frac{87\!\cdots\!00}{32\!\cdots\!39}a^{12}+\frac{12\!\cdots\!60}{10\!\cdots\!13}a^{10}+\frac{35\!\cdots\!31}{32\!\cdots\!39}a^{8}+\frac{89\!\cdots\!17}{32\!\cdots\!39}a^{6}+\frac{83\!\cdots\!97}{32\!\cdots\!39}a^{4}+\frac{17\!\cdots\!67}{10\!\cdots\!13}a^{2}-\frac{20\!\cdots\!26}{32\!\cdots\!39}$, $\frac{14\!\cdots\!24}{32\!\cdots\!39}a^{20}+\frac{14\!\cdots\!77}{32\!\cdots\!39}a^{18}-\frac{29\!\cdots\!22}{32\!\cdots\!39}a^{16}-\frac{17\!\cdots\!61}{10\!\cdots\!13}a^{14}-\frac{59\!\cdots\!24}{10\!\cdots\!13}a^{12}+\frac{87\!\cdots\!78}{32\!\cdots\!39}a^{10}+\frac{74\!\cdots\!17}{32\!\cdots\!39}a^{8}+\frac{57\!\cdots\!40}{10\!\cdots\!13}a^{6}+\frac{42\!\cdots\!40}{10\!\cdots\!13}a^{4}-\frac{66\!\cdots\!08}{10\!\cdots\!13}a^{2}-\frac{35\!\cdots\!73}{32\!\cdots\!39}$, $\frac{15\!\cdots\!16}{67\!\cdots\!99}a^{21}+\frac{49\!\cdots\!90}{20\!\cdots\!97}a^{19}-\frac{33\!\cdots\!47}{67\!\cdots\!99}a^{17}-\frac{16\!\cdots\!09}{20\!\cdots\!97}a^{15}-\frac{60\!\cdots\!02}{20\!\cdots\!97}a^{13}+\frac{26\!\cdots\!38}{20\!\cdots\!97}a^{11}+\frac{91\!\cdots\!50}{76\!\cdots\!91}a^{9}+\frac{62\!\cdots\!21}{20\!\cdots\!97}a^{7}+\frac{60\!\cdots\!63}{20\!\cdots\!97}a^{5}+\frac{10\!\cdots\!34}{20\!\cdots\!97}a^{3}-\frac{86\!\cdots\!09}{20\!\cdots\!97}a$, $\frac{17\!\cdots\!80}{67\!\cdots\!99}a^{21}-\frac{17\!\cdots\!22}{10\!\cdots\!13}a^{20}+\frac{66\!\cdots\!06}{20\!\cdots\!97}a^{19}-\frac{21\!\cdots\!64}{10\!\cdots\!13}a^{18}-\frac{33\!\cdots\!32}{67\!\cdots\!99}a^{17}+\frac{96\!\cdots\!21}{32\!\cdots\!39}a^{16}-\frac{21\!\cdots\!94}{20\!\cdots\!97}a^{15}+\frac{20\!\cdots\!37}{32\!\cdots\!39}a^{14}-\frac{10\!\cdots\!74}{20\!\cdots\!97}a^{13}+\frac{33\!\cdots\!85}{10\!\cdots\!13}a^{12}+\frac{16\!\cdots\!34}{20\!\cdots\!97}a^{11}-\frac{16\!\cdots\!96}{32\!\cdots\!39}a^{10}+\frac{12\!\cdots\!50}{76\!\cdots\!91}a^{9}-\frac{32\!\cdots\!86}{32\!\cdots\!39}a^{8}+\frac{12\!\cdots\!37}{20\!\cdots\!97}a^{7}-\frac{12\!\cdots\!14}{32\!\cdots\!39}a^{6}+\frac{21\!\cdots\!29}{20\!\cdots\!97}a^{5}-\frac{70\!\cdots\!86}{10\!\cdots\!13}a^{4}+\frac{18\!\cdots\!01}{20\!\cdots\!97}a^{3}-\frac{17\!\cdots\!39}{32\!\cdots\!39}a^{2}+\frac{58\!\cdots\!20}{20\!\cdots\!97}a-\frac{56\!\cdots\!20}{32\!\cdots\!39}$, $\frac{36\!\cdots\!97}{20\!\cdots\!97}a^{21}+\frac{78\!\cdots\!90}{10\!\cdots\!13}a^{20}+\frac{10\!\cdots\!14}{20\!\cdots\!97}a^{19}+\frac{67\!\cdots\!62}{32\!\cdots\!39}a^{18}+\frac{34\!\cdots\!15}{67\!\cdots\!99}a^{17}+\frac{22\!\cdots\!63}{10\!\cdots\!13}a^{16}+\frac{29\!\cdots\!01}{20\!\cdots\!97}a^{15}+\frac{19\!\cdots\!20}{32\!\cdots\!39}a^{14}-\frac{17\!\cdots\!86}{20\!\cdots\!97}a^{13}-\frac{11\!\cdots\!95}{32\!\cdots\!39}a^{12}-\frac{16\!\cdots\!07}{20\!\cdots\!97}a^{11}-\frac{10\!\cdots\!65}{32\!\cdots\!39}a^{10}-\frac{18\!\cdots\!81}{76\!\cdots\!91}a^{9}-\frac{10\!\cdots\!05}{10\!\cdots\!13}a^{8}-\frac{72\!\cdots\!87}{20\!\cdots\!97}a^{7}-\frac{47\!\cdots\!84}{32\!\cdots\!39}a^{6}-\frac{41\!\cdots\!56}{20\!\cdots\!97}a^{5}-\frac{27\!\cdots\!26}{32\!\cdots\!39}a^{4}+\frac{10\!\cdots\!03}{67\!\cdots\!99}a^{3}+\frac{20\!\cdots\!44}{32\!\cdots\!39}a^{2}+\frac{28\!\cdots\!97}{67\!\cdots\!99}a+\frac{55\!\cdots\!39}{32\!\cdots\!39}$, $\frac{36\!\cdots\!60}{67\!\cdots\!99}a^{21}-\frac{23\!\cdots\!38}{10\!\cdots\!13}a^{20}+\frac{10\!\cdots\!77}{67\!\cdots\!99}a^{19}-\frac{67\!\cdots\!16}{10\!\cdots\!13}a^{18}+\frac{30\!\cdots\!04}{20\!\cdots\!97}a^{17}-\frac{20\!\cdots\!71}{32\!\cdots\!39}a^{16}+\frac{87\!\cdots\!52}{20\!\cdots\!97}a^{15}-\frac{57\!\cdots\!98}{32\!\cdots\!39}a^{14}-\frac{17\!\cdots\!94}{67\!\cdots\!99}a^{13}+\frac{11\!\cdots\!27}{10\!\cdots\!13}a^{12}-\frac{48\!\cdots\!69}{20\!\cdots\!97}a^{11}+\frac{31\!\cdots\!66}{32\!\cdots\!39}a^{10}-\frac{16\!\cdots\!56}{22\!\cdots\!73}a^{9}+\frac{99\!\cdots\!48}{32\!\cdots\!39}a^{8}-\frac{21\!\cdots\!12}{20\!\cdots\!97}a^{7}+\frac{14\!\cdots\!37}{32\!\cdots\!39}a^{6}-\frac{41\!\cdots\!59}{67\!\cdots\!99}a^{5}+\frac{27\!\cdots\!67}{10\!\cdots\!13}a^{4}+\frac{94\!\cdots\!87}{20\!\cdots\!97}a^{3}-\frac{61\!\cdots\!24}{32\!\cdots\!39}a^{2}+\frac{25\!\cdots\!31}{20\!\cdots\!97}a-\frac{16\!\cdots\!88}{32\!\cdots\!39}$
| 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: | \( 585127810911000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 585127810911000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 0.707790882229475 \end{aligned}\] (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 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129)
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 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129, 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 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129);
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 - 192*x^18 - 3910*x^16 - 18357*x^14 + 37519*x^12 + 608582*x^10 + 2113799*x^8 + 3221580*x^6 + 1933376*x^4 - 120888*x^2 - 388129);
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 $C_2^{10}.\PSL(2,11)$ are not computed |
| Character table for $C_2^{10}.\PSL(2,11)$ is not computed |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.5.0.1}{5} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
|
\(74843\)
| $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |