Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1)
gp: K = bnfinit(y^21 - 5*y^20 + 11*y^19 - 5*y^18 - 29*y^17 + 64*y^16 - 22*y^15 - 117*y^14 + 218*y^13 - 124*y^12 - 76*y^11 + 132*y^10 - 38*y^9 + 31*y^8 - 168*y^7 + 204*y^6 - 47*y^5 - 120*y^4 + 134*y^3 - 63*y^2 + 13*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1)
\( x^{21} - 5 x^{20} + 11 x^{19} - 5 x^{18} - 29 x^{17} + 64 x^{16} - 22 x^{15} - 117 x^{14} + 218 x^{13} + \cdots - 1 \)
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: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-336444035684435024524918784\)
\(\medspace = -\,2^{14}\cdot 11^{13}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.33\) | 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: | $2^{2/3}11^{5/6}29^{2/3}\approx 110.52127846308204$ | ||
| Ramified primes: |
\(2\), \(11\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{620672130980947}a^{20}-\frac{269511794070246}{620672130980947}a^{19}-\frac{255523082199229}{620672130980947}a^{18}-\frac{218038955154873}{620672130980947}a^{17}-\frac{37351224254539}{620672130980947}a^{16}-\frac{144211209449026}{620672130980947}a^{15}+\frac{269033033665895}{620672130980947}a^{14}-\frac{188904534648443}{620672130980947}a^{13}+\frac{218952732589152}{620672130980947}a^{12}+\frac{51254447120798}{620672130980947}a^{11}+\frac{59784868402271}{620672130980947}a^{10}-\frac{154218060155713}{620672130980947}a^{9}-\frac{15877221108758}{620672130980947}a^{8}-\frac{262682960946493}{620672130980947}a^{7}-\frac{275420565823400}{620672130980947}a^{6}-\frac{216367859012951}{620672130980947}a^{5}+\frac{47543204046642}{620672130980947}a^{4}+\frac{183292206343283}{620672130980947}a^{3}-\frac{174493987896416}{620672130980947}a^{2}-\frac{970412315125}{620672130980947}a+\frac{220506429574840}{620672130980947}$
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: | $11$ | 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: |
$\frac{321255766212698}{620672130980947}a^{20}-\frac{13\!\cdots\!92}{620672130980947}a^{19}+\frac{22\!\cdots\!68}{620672130980947}a^{18}+\frac{868460879224850}{620672130980947}a^{17}-\frac{94\!\cdots\!90}{620672130980947}a^{16}+\frac{12\!\cdots\!44}{620672130980947}a^{15}+\frac{60\!\cdots\!04}{620672130980947}a^{14}-\frac{36\!\cdots\!84}{620672130980947}a^{13}+\frac{39\!\cdots\!37}{620672130980947}a^{12}+\frac{18\!\cdots\!80}{620672130980947}a^{11}-\frac{36\!\cdots\!66}{620672130980947}a^{10}+\frac{21\!\cdots\!26}{620672130980947}a^{9}+\frac{13\!\cdots\!82}{620672130980947}a^{8}+\frac{14\!\cdots\!16}{620672130980947}a^{7}-\frac{44\!\cdots\!68}{620672130980947}a^{6}+\frac{25\!\cdots\!88}{620672130980947}a^{5}+\frac{17\!\cdots\!28}{620672130980947}a^{4}-\frac{36\!\cdots\!18}{620672130980947}a^{3}+\frac{20\!\cdots\!46}{620672130980947}a^{2}-\frac{45\!\cdots\!68}{620672130980947}a-\frac{406961633641940}{620672130980947}$, $\frac{937101279976823}{620672130980947}a^{20}-\frac{43\!\cdots\!20}{620672130980947}a^{19}+\frac{89\!\cdots\!98}{620672130980947}a^{18}-\frac{21\!\cdots\!05}{620672130980947}a^{17}-\frac{27\!\cdots\!86}{620672130980947}a^{16}+\frac{51\!\cdots\!17}{620672130980947}a^{15}-\frac{62\!\cdots\!78}{620672130980947}a^{14}-\frac{10\!\cdots\!09}{620672130980947}a^{13}+\frac{17\!\cdots\!24}{620672130980947}a^{12}-\frac{69\!\cdots\!40}{620672130980947}a^{11}-\frac{84\!\cdots\!24}{620672130980947}a^{10}+\frac{96\!\cdots\!16}{620672130980947}a^{9}-\frac{12\!\cdots\!52}{620672130980947}a^{8}+\frac{30\!\cdots\!21}{620672130980947}a^{7}-\frac{14\!\cdots\!13}{620672130980947}a^{6}+\frac{14\!\cdots\!43}{620672130980947}a^{5}-\frac{75\!\cdots\!16}{620672130980947}a^{4}-\frac{10\!\cdots\!77}{620672130980947}a^{3}+\frac{94\!\cdots\!03}{620672130980947}a^{2}-\frac{36\!\cdots\!95}{620672130980947}a+\frac{39\!\cdots\!08}{620672130980947}$, $a$, $\frac{669850233115805}{620672130980947}a^{20}-\frac{32\!\cdots\!82}{620672130980947}a^{19}+\frac{68\!\cdots\!36}{620672130980947}a^{18}-\frac{17\!\cdots\!54}{620672130980947}a^{17}-\frac{21\!\cdots\!24}{620672130980947}a^{16}+\frac{41\!\cdots\!28}{620672130980947}a^{15}-\frac{60\!\cdots\!50}{620672130980947}a^{14}-\frac{88\!\cdots\!00}{620672130980947}a^{13}+\frac{13\!\cdots\!39}{620672130980947}a^{12}-\frac{52\!\cdots\!93}{620672130980947}a^{11}-\frac{85\!\cdots\!06}{620672130980947}a^{10}+\frac{10\!\cdots\!29}{620672130980947}a^{9}-\frac{20\!\cdots\!54}{620672130980947}a^{8}+\frac{23\!\cdots\!59}{620672130980947}a^{7}-\frac{11\!\cdots\!48}{620672130980947}a^{6}+\frac{12\!\cdots\!49}{620672130980947}a^{5}+\frac{36\!\cdots\!83}{620672130980947}a^{4}-\frac{10\!\cdots\!63}{620672130980947}a^{3}+\frac{93\!\cdots\!86}{620672130980947}a^{2}-\frac{34\!\cdots\!30}{620672130980947}a+\frac{41\!\cdots\!78}{620672130980947}$, $\frac{13\!\cdots\!40}{620672130980947}a^{20}-\frac{62\!\cdots\!03}{620672130980947}a^{19}+\frac{12\!\cdots\!26}{620672130980947}a^{18}-\frac{25\!\cdots\!18}{620672130980947}a^{17}-\frac{39\!\cdots\!46}{620672130980947}a^{16}+\frac{72\!\cdots\!11}{620672130980947}a^{15}-\frac{52\!\cdots\!74}{620672130980947}a^{14}-\frac{15\!\cdots\!37}{620672130980947}a^{13}+\frac{23\!\cdots\!82}{620672130980947}a^{12}-\frac{86\!\cdots\!87}{620672130980947}a^{11}-\frac{12\!\cdots\!91}{620672130980947}a^{10}+\frac{13\!\cdots\!55}{620672130980947}a^{9}-\frac{55\!\cdots\!22}{620672130980947}a^{8}+\frac{41\!\cdots\!09}{620672130980947}a^{7}-\frac{21\!\cdots\!04}{620672130980947}a^{6}+\frac{20\!\cdots\!03}{620672130980947}a^{5}+\frac{55\!\cdots\!25}{620672130980947}a^{4}-\frac{15\!\cdots\!80}{620672130980947}a^{3}+\frac{12\!\cdots\!94}{620672130980947}a^{2}-\frac{42\!\cdots\!69}{620672130980947}a+\frac{49\!\cdots\!01}{620672130980947}$, $\frac{321255766212698}{620672130980947}a^{20}-\frac{13\!\cdots\!92}{620672130980947}a^{19}+\frac{22\!\cdots\!68}{620672130980947}a^{18}+\frac{868460879224850}{620672130980947}a^{17}-\frac{94\!\cdots\!90}{620672130980947}a^{16}+\frac{12\!\cdots\!44}{620672130980947}a^{15}+\frac{60\!\cdots\!04}{620672130980947}a^{14}-\frac{36\!\cdots\!84}{620672130980947}a^{13}+\frac{39\!\cdots\!37}{620672130980947}a^{12}+\frac{18\!\cdots\!80}{620672130980947}a^{11}-\frac{36\!\cdots\!66}{620672130980947}a^{10}+\frac{21\!\cdots\!26}{620672130980947}a^{9}+\frac{13\!\cdots\!82}{620672130980947}a^{8}+\frac{14\!\cdots\!16}{620672130980947}a^{7}-\frac{44\!\cdots\!68}{620672130980947}a^{6}+\frac{25\!\cdots\!88}{620672130980947}a^{5}+\frac{17\!\cdots\!28}{620672130980947}a^{4}-\frac{36\!\cdots\!18}{620672130980947}a^{3}+\frac{20\!\cdots\!46}{620672130980947}a^{2}-\frac{51\!\cdots\!15}{620672130980947}a-\frac{406961633641940}{620672130980947}$, $\frac{10\!\cdots\!59}{620672130980947}a^{20}-\frac{51\!\cdots\!13}{620672130980947}a^{19}+\frac{10\!\cdots\!03}{620672130980947}a^{18}-\frac{27\!\cdots\!39}{620672130980947}a^{17}-\frac{31\!\cdots\!25}{620672130980947}a^{16}+\frac{60\!\cdots\!45}{620672130980947}a^{15}-\frac{82\!\cdots\!96}{620672130980947}a^{14}-\frac{12\!\cdots\!27}{620672130980947}a^{13}+\frac{19\!\cdots\!12}{620672130980947}a^{12}-\frac{84\!\cdots\!84}{620672130980947}a^{11}-\frac{93\!\cdots\!19}{620672130980947}a^{10}+\frac{10\!\cdots\!51}{620672130980947}a^{9}-\frac{12\!\cdots\!94}{620672130980947}a^{8}+\frac{33\!\cdots\!90}{620672130980947}a^{7}-\frac{16\!\cdots\!88}{620672130980947}a^{6}+\frac{17\!\cdots\!01}{620672130980947}a^{5}-\frac{12\!\cdots\!31}{620672130980947}a^{4}-\frac{12\!\cdots\!86}{620672130980947}a^{3}+\frac{10\!\cdots\!47}{620672130980947}a^{2}-\frac{39\!\cdots\!64}{620672130980947}a+\frac{39\!\cdots\!38}{620672130980947}$, $\frac{546367970247636}{620672130980947}a^{20}-\frac{27\!\cdots\!95}{620672130980947}a^{19}+\frac{61\!\cdots\!70}{620672130980947}a^{18}-\frac{31\!\cdots\!43}{620672130980947}a^{17}-\frac{15\!\cdots\!54}{620672130980947}a^{16}+\frac{35\!\cdots\!86}{620672130980947}a^{15}-\frac{13\!\cdots\!95}{620672130980947}a^{14}-\frac{63\!\cdots\!50}{620672130980947}a^{13}+\frac{12\!\cdots\!69}{620672130980947}a^{12}-\frac{72\!\cdots\!44}{620672130980947}a^{11}-\frac{37\!\cdots\!12}{620672130980947}a^{10}+\frac{67\!\cdots\!54}{620672130980947}a^{9}-\frac{14\!\cdots\!11}{620672130980947}a^{8}+\frac{10\!\cdots\!60}{620672130980947}a^{7}-\frac{90\!\cdots\!17}{620672130980947}a^{6}+\frac{11\!\cdots\!06}{620672130980947}a^{5}-\frac{29\!\cdots\!68}{620672130980947}a^{4}-\frac{62\!\cdots\!37}{620672130980947}a^{3}+\frac{67\!\cdots\!93}{620672130980947}a^{2}-\frac{30\!\cdots\!55}{620672130980947}a+\frac{42\!\cdots\!79}{620672130980947}$, $\frac{658896252815737}{620672130980947}a^{20}-\frac{29\!\cdots\!96}{620672130980947}a^{19}+\frac{56\!\cdots\!01}{620672130980947}a^{18}-\frac{342305025900542}{620672130980947}a^{17}-\frac{18\!\cdots\!22}{620672130980947}a^{16}+\frac{31\!\cdots\!59}{620672130980947}a^{15}+\frac{16\!\cdots\!12}{620672130980947}a^{14}-\frac{72\!\cdots\!62}{620672130980947}a^{13}+\frac{10\!\cdots\!17}{620672130980947}a^{12}-\frac{31\!\cdots\!78}{620672130980947}a^{11}-\frac{54\!\cdots\!73}{620672130980947}a^{10}+\frac{50\!\cdots\!65}{620672130980947}a^{9}-\frac{48\!\cdots\!27}{620672130980947}a^{8}+\frac{25\!\cdots\!08}{620672130980947}a^{7}-\frac{93\!\cdots\!15}{620672130980947}a^{6}+\frac{80\!\cdots\!44}{620672130980947}a^{5}+\frac{33\!\cdots\!48}{620672130980947}a^{4}-\frac{64\!\cdots\!91}{620672130980947}a^{3}+\frac{52\!\cdots\!25}{620672130980947}a^{2}-\frac{20\!\cdots\!55}{620672130980947}a+\frac{26\!\cdots\!45}{620672130980947}$, $\frac{10\!\cdots\!30}{620672130980947}a^{20}-\frac{51\!\cdots\!90}{620672130980947}a^{19}+\frac{10\!\cdots\!70}{620672130980947}a^{18}-\frac{30\!\cdots\!46}{620672130980947}a^{17}-\frac{31\!\cdots\!31}{620672130980947}a^{16}+\frac{61\!\cdots\!89}{620672130980947}a^{15}-\frac{99\!\cdots\!64}{620672130980947}a^{14}-\frac{12\!\cdots\!97}{620672130980947}a^{13}+\frac{20\!\cdots\!33}{620672130980947}a^{12}-\frac{87\!\cdots\!05}{620672130980947}a^{11}-\frac{96\!\cdots\!05}{620672130980947}a^{10}+\frac{11\!\cdots\!31}{620672130980947}a^{9}-\frac{14\!\cdots\!65}{620672130980947}a^{8}+\frac{32\!\cdots\!59}{620672130980947}a^{7}-\frac{17\!\cdots\!10}{620672130980947}a^{6}+\frac{17\!\cdots\!23}{620672130980947}a^{5}-\frac{12\!\cdots\!03}{620672130980947}a^{4}-\frac{12\!\cdots\!87}{620672130980947}a^{3}+\frac{11\!\cdots\!56}{620672130980947}a^{2}-\frac{43\!\cdots\!44}{620672130980947}a+\frac{58\!\cdots\!30}{620672130980947}$, $\frac{606988014351792}{620672130980947}a^{20}-\frac{29\!\cdots\!06}{620672130980947}a^{19}+\frac{62\!\cdots\!95}{620672130980947}a^{18}-\frac{18\!\cdots\!44}{620672130980947}a^{17}-\frac{19\!\cdots\!33}{620672130980947}a^{16}+\frac{37\!\cdots\!97}{620672130980947}a^{15}-\frac{58\!\cdots\!04}{620672130980947}a^{14}-\frac{78\!\cdots\!71}{620672130980947}a^{13}+\frac{12\!\cdots\!38}{620672130980947}a^{12}-\frac{48\!\cdots\!88}{620672130980947}a^{11}-\frac{70\!\cdots\!47}{620672130980947}a^{10}+\frac{79\!\cdots\!98}{620672130980947}a^{9}-\frac{89\!\cdots\!35}{620672130980947}a^{8}+\frac{18\!\cdots\!43}{620672130980947}a^{7}-\frac{10\!\cdots\!03}{620672130980947}a^{6}+\frac{10\!\cdots\!73}{620672130980947}a^{5}+\frac{25\!\cdots\!19}{620672130980947}a^{4}-\frac{86\!\cdots\!10}{620672130980947}a^{3}+\frac{72\!\cdots\!60}{620672130980947}a^{2}-\frac{25\!\cdots\!35}{620672130980947}a+\frac{31\!\cdots\!83}{620672130980947}$
| 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: | \( 73219.6797146 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 73219.6797146 \cdot 1}{2\cdot\sqrt{336444035684435024524918784}}\cr\approx \mathstrut & 0.243697017730 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1)
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 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1, 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 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1);
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 - 5*x^20 + 11*x^19 - 5*x^18 - 29*x^17 + 64*x^16 - 22*x^15 - 117*x^14 + 218*x^13 - 124*x^12 - 76*x^11 + 132*x^10 - 38*x^9 + 31*x^8 - 168*x^7 + 204*x^6 - 47*x^5 - 120*x^4 + 134*x^3 - 63*x^2 + 13*x - 1);
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
$S_3\times A_7$ (as 21T57):
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 15120 |
| The 27 conjugacy class representatives for $S_3\times A_7$ |
| Character table for $S_3\times A_7$ |
Intermediate fields
| 3.1.44.1, 7.3.12313081.1 |
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 42 sibling: | data not computed |
| Degree 45 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 | $21$ | $15{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | $21$ | R | $21$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.7.0.1}{7} }^{3}$ | $15{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.14.1 | $x^{21} + 14 x^{18} + 87 x^{15} + 3 x^{14} - 14 x^{12} - 462 x^{11} + 1655 x^{9} + 4290 x^{8} + 3 x^{7} + 2982 x^{6} - 6090 x^{5} + 210 x^{4} - 1651 x^{3} + 1263 x^{2} + 87 x + 251$ | $3$ | $7$ | $14$ | 21T6 | $[\ ]_{3}^{14}$ |
|
\(11\)
| 11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 11.6.5.2 | $x^{6} + 11$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.3.2.1 | $x^{3} + 29$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.6.4.1 | $x^{6} + 72 x^{5} + 1734 x^{4} + 14170 x^{3} + 5556 x^{2} + 50052 x + 397569$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |