Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4)
gp: K = bnfinit(y^21 - 2*y^20 - 34*y^19 + 44*y^18 + 440*y^17 - 336*y^16 - 2735*y^15 + 1258*y^14 + 9114*y^13 - 2548*y^12 - 17108*y^11 + 2708*y^10 + 18395*y^9 - 1186*y^8 - 11144*y^7 - 160*y^6 + 3584*y^5 + 284*y^4 - 539*y^3 - 66*y^2 + 28*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4)
\( x^{21} - 2 x^{20} - 34 x^{19} + 44 x^{18} + 440 x^{17} - 336 x^{16} - 2735 x^{15} + 1258 x^{14} + \cdots + 4 \)
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: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(114304429232803182734215052479732645888\)
\(\medspace = 2^{32}\cdot 37^{7}\cdot 809^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.91\) | 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: | $2^{13/6}37^{1/2}809^{1/2}\approx 776.795641455365$ | ||
| Ramified primes: |
\(2\), \(37\), \(809\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{25\!\cdots\!78}a^{20}+\frac{77\!\cdots\!63}{12\!\cdots\!39}a^{19}+\frac{19\!\cdots\!72}{12\!\cdots\!39}a^{18}-\frac{42\!\cdots\!37}{12\!\cdots\!39}a^{17}+\frac{32\!\cdots\!02}{12\!\cdots\!39}a^{16}-\frac{13\!\cdots\!43}{12\!\cdots\!39}a^{15}-\frac{58\!\cdots\!11}{25\!\cdots\!78}a^{14}+\frac{21\!\cdots\!94}{12\!\cdots\!39}a^{13}-\frac{40\!\cdots\!23}{12\!\cdots\!39}a^{12}-\frac{12\!\cdots\!68}{12\!\cdots\!39}a^{11}+\frac{45\!\cdots\!87}{12\!\cdots\!39}a^{10}+\frac{14\!\cdots\!69}{12\!\cdots\!39}a^{9}-\frac{12\!\cdots\!27}{25\!\cdots\!78}a^{8}+\frac{61\!\cdots\!48}{12\!\cdots\!39}a^{7}+\frac{54\!\cdots\!20}{12\!\cdots\!39}a^{6}+\frac{38\!\cdots\!66}{12\!\cdots\!39}a^{5}-\frac{30\!\cdots\!32}{12\!\cdots\!39}a^{4}+\frac{19\!\cdots\!75}{12\!\cdots\!39}a^{3}+\frac{49\!\cdots\!29}{25\!\cdots\!78}a^{2}-\frac{34\!\cdots\!54}{12\!\cdots\!39}a-\frac{40\!\cdots\!90}{12\!\cdots\!39}$
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: | $20$ | 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: |
$\frac{44\!\cdots\!33}{25\!\cdots\!78}a^{20}-\frac{53\!\cdots\!67}{12\!\cdots\!39}a^{19}-\frac{72\!\cdots\!86}{12\!\cdots\!39}a^{18}+\frac{12\!\cdots\!80}{12\!\cdots\!39}a^{17}+\frac{91\!\cdots\!13}{12\!\cdots\!39}a^{16}-\frac{10\!\cdots\!02}{12\!\cdots\!39}a^{15}-\frac{10\!\cdots\!97}{25\!\cdots\!78}a^{14}+\frac{48\!\cdots\!77}{12\!\cdots\!39}a^{13}+\frac{17\!\cdots\!75}{12\!\cdots\!39}a^{12}-\frac{12\!\cdots\!30}{12\!\cdots\!39}a^{11}-\frac{30\!\cdots\!40}{12\!\cdots\!39}a^{10}+\frac{17\!\cdots\!14}{12\!\cdots\!39}a^{9}+\frac{57\!\cdots\!69}{25\!\cdots\!78}a^{8}-\frac{13\!\cdots\!03}{12\!\cdots\!39}a^{7}-\frac{14\!\cdots\!35}{12\!\cdots\!39}a^{6}+\frac{54\!\cdots\!36}{12\!\cdots\!39}a^{5}+\frac{36\!\cdots\!45}{12\!\cdots\!39}a^{4}-\frac{91\!\cdots\!36}{12\!\cdots\!39}a^{3}-\frac{76\!\cdots\!59}{25\!\cdots\!78}a^{2}+\frac{43\!\cdots\!07}{12\!\cdots\!39}a+\frac{12\!\cdots\!71}{12\!\cdots\!39}$, $\frac{37\!\cdots\!20}{12\!\cdots\!39}a^{20}-\frac{96\!\cdots\!55}{12\!\cdots\!39}a^{19}-\frac{12\!\cdots\!78}{12\!\cdots\!39}a^{18}+\frac{23\!\cdots\!08}{12\!\cdots\!39}a^{17}+\frac{15\!\cdots\!26}{12\!\cdots\!39}a^{16}-\frac{21\!\cdots\!11}{12\!\cdots\!39}a^{15}-\frac{90\!\cdots\!38}{12\!\cdots\!39}a^{14}+\frac{10\!\cdots\!74}{12\!\cdots\!39}a^{13}+\frac{28\!\cdots\!47}{12\!\cdots\!39}a^{12}-\frac{26\!\cdots\!11}{12\!\cdots\!39}a^{11}-\frac{49\!\cdots\!28}{12\!\cdots\!39}a^{10}+\frac{39\!\cdots\!55}{12\!\cdots\!39}a^{9}+\frac{48\!\cdots\!92}{12\!\cdots\!39}a^{8}-\frac{32\!\cdots\!97}{12\!\cdots\!39}a^{7}-\frac{24\!\cdots\!32}{12\!\cdots\!39}a^{6}+\frac{13\!\cdots\!05}{12\!\cdots\!39}a^{5}+\frac{62\!\cdots\!06}{12\!\cdots\!39}a^{4}-\frac{25\!\cdots\!03}{12\!\cdots\!39}a^{3}-\frac{66\!\cdots\!90}{12\!\cdots\!39}a^{2}+\frac{14\!\cdots\!36}{12\!\cdots\!39}a+\frac{25\!\cdots\!09}{12\!\cdots\!39}$, $\frac{10\!\cdots\!50}{12\!\cdots\!39}a^{20}-\frac{27\!\cdots\!47}{12\!\cdots\!39}a^{19}-\frac{34\!\cdots\!42}{12\!\cdots\!39}a^{18}+\frac{66\!\cdots\!08}{12\!\cdots\!39}a^{17}+\frac{42\!\cdots\!97}{12\!\cdots\!39}a^{16}-\frac{60\!\cdots\!01}{12\!\cdots\!39}a^{15}-\frac{25\!\cdots\!00}{12\!\cdots\!39}a^{14}+\frac{28\!\cdots\!74}{12\!\cdots\!39}a^{13}+\frac{80\!\cdots\!82}{12\!\cdots\!39}a^{12}-\frac{74\!\cdots\!99}{12\!\cdots\!39}a^{11}-\frac{14\!\cdots\!16}{12\!\cdots\!39}a^{10}+\frac{11\!\cdots\!47}{12\!\cdots\!39}a^{9}+\frac{13\!\cdots\!76}{12\!\cdots\!39}a^{8}-\frac{96\!\cdots\!21}{12\!\cdots\!39}a^{7}-\frac{70\!\cdots\!17}{12\!\cdots\!39}a^{6}+\frac{42\!\cdots\!97}{12\!\cdots\!39}a^{5}+\frac{18\!\cdots\!89}{12\!\cdots\!39}a^{4}-\frac{87\!\cdots\!29}{12\!\cdots\!39}a^{3}-\frac{24\!\cdots\!34}{12\!\cdots\!39}a^{2}+\frac{56\!\cdots\!24}{12\!\cdots\!39}a+\frac{10\!\cdots\!89}{12\!\cdots\!39}$, $\frac{36\!\cdots\!72}{12\!\cdots\!39}a^{20}-\frac{93\!\cdots\!00}{12\!\cdots\!39}a^{19}-\frac{11\!\cdots\!70}{12\!\cdots\!39}a^{18}+\frac{22\!\cdots\!50}{12\!\cdots\!39}a^{17}+\frac{14\!\cdots\!65}{12\!\cdots\!39}a^{16}-\frac{20\!\cdots\!69}{12\!\cdots\!39}a^{15}-\frac{89\!\cdots\!48}{12\!\cdots\!39}a^{14}+\frac{96\!\cdots\!92}{12\!\cdots\!39}a^{13}+\frac{28\!\cdots\!81}{12\!\cdots\!39}a^{12}-\frac{25\!\cdots\!20}{12\!\cdots\!39}a^{11}-\frac{50\!\cdots\!40}{12\!\cdots\!39}a^{10}+\frac{38\!\cdots\!26}{12\!\cdots\!39}a^{9}+\frac{49\!\cdots\!38}{12\!\cdots\!39}a^{8}-\frac{31\!\cdots\!96}{12\!\cdots\!39}a^{7}-\frac{26\!\cdots\!17}{12\!\cdots\!39}a^{6}+\frac{13\!\cdots\!88}{12\!\cdots\!39}a^{5}+\frac{69\!\cdots\!93}{12\!\cdots\!39}a^{4}-\frac{26\!\cdots\!53}{12\!\cdots\!39}a^{3}-\frac{82\!\cdots\!82}{12\!\cdots\!39}a^{2}+\frac{16\!\cdots\!40}{12\!\cdots\!39}a+\frac{34\!\cdots\!55}{12\!\cdots\!39}$, $\frac{19\!\cdots\!38}{12\!\cdots\!39}a^{20}-\frac{50\!\cdots\!71}{12\!\cdots\!39}a^{19}-\frac{61\!\cdots\!48}{12\!\cdots\!39}a^{18}+\frac{12\!\cdots\!54}{12\!\cdots\!39}a^{17}+\frac{76\!\cdots\!79}{12\!\cdots\!39}a^{16}-\frac{11\!\cdots\!64}{12\!\cdots\!39}a^{15}-\frac{45\!\cdots\!08}{12\!\cdots\!39}a^{14}+\frac{55\!\cdots\!57}{12\!\cdots\!39}a^{13}+\frac{14\!\cdots\!32}{12\!\cdots\!39}a^{12}-\frac{14\!\cdots\!18}{12\!\cdots\!39}a^{11}-\frac{25\!\cdots\!84}{12\!\cdots\!39}a^{10}+\frac{22\!\cdots\!44}{12\!\cdots\!39}a^{9}+\frac{24\!\cdots\!50}{12\!\cdots\!39}a^{8}-\frac{18\!\cdots\!33}{12\!\cdots\!39}a^{7}-\frac{12\!\cdots\!74}{12\!\cdots\!39}a^{6}+\frac{81\!\cdots\!56}{12\!\cdots\!39}a^{5}+\frac{32\!\cdots\!17}{12\!\cdots\!39}a^{4}-\frac{15\!\cdots\!04}{12\!\cdots\!39}a^{3}-\frac{39\!\cdots\!12}{12\!\cdots\!39}a^{2}+\frac{10\!\cdots\!13}{12\!\cdots\!39}a+\frac{17\!\cdots\!99}{12\!\cdots\!39}$, $\frac{11\!\cdots\!98}{12\!\cdots\!39}a^{20}-\frac{30\!\cdots\!02}{12\!\cdots\!39}a^{19}-\frac{35\!\cdots\!50}{12\!\cdots\!39}a^{18}+\frac{74\!\cdots\!66}{12\!\cdots\!39}a^{17}+\frac{44\!\cdots\!58}{12\!\cdots\!39}a^{16}-\frac{68\!\cdots\!43}{12\!\cdots\!39}a^{15}-\frac{25\!\cdots\!90}{12\!\cdots\!39}a^{14}+\frac{32\!\cdots\!56}{12\!\cdots\!39}a^{13}+\frac{80\!\cdots\!48}{12\!\cdots\!39}a^{12}-\frac{86\!\cdots\!90}{12\!\cdots\!39}a^{11}-\frac{13\!\cdots\!04}{12\!\cdots\!39}a^{10}+\frac{12\!\cdots\!76}{12\!\cdots\!39}a^{9}+\frac{12\!\cdots\!30}{12\!\cdots\!39}a^{8}-\frac{10\!\cdots\!22}{12\!\cdots\!39}a^{7}-\frac{56\!\cdots\!32}{12\!\cdots\!39}a^{6}+\frac{43\!\cdots\!14}{12\!\cdots\!39}a^{5}+\frac{11\!\cdots\!02}{12\!\cdots\!39}a^{4}-\frac{73\!\cdots\!79}{12\!\cdots\!39}a^{3}-\frac{84\!\cdots\!42}{12\!\cdots\!39}a^{2}+\frac{31\!\cdots\!20}{12\!\cdots\!39}a+\frac{18\!\cdots\!43}{12\!\cdots\!39}$, $\frac{19\!\cdots\!57}{12\!\cdots\!39}a^{20}-\frac{50\!\cdots\!81}{12\!\cdots\!39}a^{19}-\frac{64\!\cdots\!57}{12\!\cdots\!39}a^{18}+\frac{12\!\cdots\!61}{12\!\cdots\!39}a^{17}+\frac{80\!\cdots\!67}{12\!\cdots\!39}a^{16}-\frac{10\!\cdots\!65}{12\!\cdots\!39}a^{15}-\frac{48\!\cdots\!14}{12\!\cdots\!39}a^{14}+\frac{50\!\cdots\!50}{12\!\cdots\!39}a^{13}+\frac{15\!\cdots\!30}{12\!\cdots\!39}a^{12}-\frac{13\!\cdots\!29}{12\!\cdots\!39}a^{11}-\frac{26\!\cdots\!97}{12\!\cdots\!39}a^{10}+\frac{19\!\cdots\!05}{12\!\cdots\!39}a^{9}+\frac{25\!\cdots\!10}{12\!\cdots\!39}a^{8}-\frac{15\!\cdots\!58}{12\!\cdots\!39}a^{7}-\frac{13\!\cdots\!75}{12\!\cdots\!39}a^{6}+\frac{66\!\cdots\!55}{12\!\cdots\!39}a^{5}+\frac{33\!\cdots\!09}{12\!\cdots\!39}a^{4}-\frac{12\!\cdots\!77}{12\!\cdots\!39}a^{3}-\frac{37\!\cdots\!04}{12\!\cdots\!39}a^{2}+\frac{72\!\cdots\!67}{12\!\cdots\!39}a+\frac{14\!\cdots\!51}{12\!\cdots\!39}$, $\frac{10\!\cdots\!13}{25\!\cdots\!78}a^{20}-\frac{11\!\cdots\!00}{12\!\cdots\!39}a^{19}-\frac{16\!\cdots\!22}{12\!\cdots\!39}a^{18}+\frac{27\!\cdots\!02}{12\!\cdots\!39}a^{17}+\frac{21\!\cdots\!40}{12\!\cdots\!39}a^{16}-\frac{23\!\cdots\!66}{12\!\cdots\!39}a^{15}-\frac{25\!\cdots\!23}{25\!\cdots\!78}a^{14}+\frac{10\!\cdots\!14}{12\!\cdots\!39}a^{13}+\frac{39\!\cdots\!53}{12\!\cdots\!39}a^{12}-\frac{24\!\cdots\!14}{12\!\cdots\!39}a^{11}-\frac{68\!\cdots\!89}{12\!\cdots\!39}a^{10}+\frac{34\!\cdots\!14}{12\!\cdots\!39}a^{9}+\frac{12\!\cdots\!67}{25\!\cdots\!78}a^{8}-\frac{27\!\cdots\!00}{12\!\cdots\!39}a^{7}-\frac{30\!\cdots\!86}{12\!\cdots\!39}a^{6}+\frac{11\!\cdots\!94}{12\!\cdots\!39}a^{5}+\frac{70\!\cdots\!49}{12\!\cdots\!39}a^{4}-\frac{20\!\cdots\!14}{12\!\cdots\!39}a^{3}-\frac{13\!\cdots\!13}{25\!\cdots\!78}a^{2}+\frac{13\!\cdots\!22}{12\!\cdots\!39}a+\frac{26\!\cdots\!63}{12\!\cdots\!39}$, $\frac{79\!\cdots\!31}{12\!\cdots\!39}a^{20}-\frac{15\!\cdots\!40}{12\!\cdots\!39}a^{19}-\frac{27\!\cdots\!39}{12\!\cdots\!39}a^{18}+\frac{32\!\cdots\!45}{12\!\cdots\!39}a^{17}+\frac{35\!\cdots\!69}{12\!\cdots\!39}a^{16}-\frac{24\!\cdots\!91}{12\!\cdots\!39}a^{15}-\frac{22\!\cdots\!52}{12\!\cdots\!39}a^{14}+\frac{91\!\cdots\!26}{12\!\cdots\!39}a^{13}+\frac{74\!\cdots\!73}{12\!\cdots\!39}a^{12}-\frac{20\!\cdots\!97}{12\!\cdots\!39}a^{11}-\frac{14\!\cdots\!23}{12\!\cdots\!39}a^{10}+\frac{28\!\cdots\!52}{12\!\cdots\!39}a^{9}+\frac{15\!\cdots\!94}{12\!\cdots\!39}a^{8}-\frac{24\!\cdots\!38}{12\!\cdots\!39}a^{7}-\frac{94\!\cdots\!82}{12\!\cdots\!39}a^{6}+\frac{13\!\cdots\!39}{12\!\cdots\!39}a^{5}+\frac{29\!\cdots\!10}{12\!\cdots\!39}a^{4}-\frac{40\!\cdots\!04}{12\!\cdots\!39}a^{3}-\frac{40\!\cdots\!86}{12\!\cdots\!39}a^{2}+\frac{50\!\cdots\!81}{12\!\cdots\!39}a+\frac{12\!\cdots\!99}{12\!\cdots\!39}$, $\frac{24\!\cdots\!28}{12\!\cdots\!39}a^{20}-\frac{83\!\cdots\!40}{12\!\cdots\!39}a^{19}-\frac{72\!\cdots\!66}{12\!\cdots\!39}a^{18}+\frac{21\!\cdots\!53}{12\!\cdots\!39}a^{17}+\frac{78\!\cdots\!14}{12\!\cdots\!39}a^{16}-\frac{21\!\cdots\!46}{12\!\cdots\!39}a^{15}-\frac{37\!\cdots\!34}{12\!\cdots\!39}a^{14}+\frac{10\!\cdots\!94}{12\!\cdots\!39}a^{13}+\frac{77\!\cdots\!67}{12\!\cdots\!39}a^{12}-\frac{29\!\cdots\!70}{12\!\cdots\!39}a^{11}-\frac{23\!\cdots\!88}{12\!\cdots\!39}a^{10}+\frac{45\!\cdots\!92}{12\!\cdots\!39}a^{9}-\frac{13\!\cdots\!78}{12\!\cdots\!39}a^{8}-\frac{39\!\cdots\!83}{12\!\cdots\!39}a^{7}+\frac{17\!\cdots\!98}{12\!\cdots\!39}a^{6}+\frac{17\!\cdots\!51}{12\!\cdots\!39}a^{5}-\frac{77\!\cdots\!08}{12\!\cdots\!39}a^{4}-\frac{33\!\cdots\!80}{12\!\cdots\!39}a^{3}+\frac{10\!\cdots\!00}{12\!\cdots\!39}a^{2}+\frac{14\!\cdots\!42}{12\!\cdots\!39}a-\frac{29\!\cdots\!45}{12\!\cdots\!39}$, $\frac{68\!\cdots\!35}{25\!\cdots\!78}a^{20}-\frac{87\!\cdots\!65}{12\!\cdots\!39}a^{19}-\frac{11\!\cdots\!06}{12\!\cdots\!39}a^{18}+\frac{21\!\cdots\!88}{12\!\cdots\!39}a^{17}+\frac{13\!\cdots\!47}{12\!\cdots\!39}a^{16}-\frac{19\!\cdots\!61}{12\!\cdots\!39}a^{15}-\frac{16\!\cdots\!27}{25\!\cdots\!78}a^{14}+\frac{88\!\cdots\!54}{12\!\cdots\!39}a^{13}+\frac{26\!\cdots\!29}{12\!\cdots\!39}a^{12}-\frac{23\!\cdots\!52}{12\!\cdots\!39}a^{11}-\frac{45\!\cdots\!19}{12\!\cdots\!39}a^{10}+\frac{34\!\cdots\!67}{12\!\cdots\!39}a^{9}+\frac{88\!\cdots\!43}{25\!\cdots\!78}a^{8}-\frac{28\!\cdots\!97}{12\!\cdots\!39}a^{7}-\frac{22\!\cdots\!28}{12\!\cdots\!39}a^{6}+\frac{11\!\cdots\!93}{12\!\cdots\!39}a^{5}+\frac{57\!\cdots\!87}{12\!\cdots\!39}a^{4}-\frac{21\!\cdots\!21}{12\!\cdots\!39}a^{3}-\frac{12\!\cdots\!51}{25\!\cdots\!78}a^{2}+\frac{12\!\cdots\!59}{12\!\cdots\!39}a+\frac{23\!\cdots\!21}{12\!\cdots\!39}$, $\frac{18\!\cdots\!85}{25\!\cdots\!78}a^{20}-\frac{23\!\cdots\!74}{12\!\cdots\!39}a^{19}-\frac{30\!\cdots\!63}{12\!\cdots\!39}a^{18}+\frac{56\!\cdots\!93}{12\!\cdots\!39}a^{17}+\frac{38\!\cdots\!10}{12\!\cdots\!39}a^{16}-\frac{50\!\cdots\!81}{12\!\cdots\!39}a^{15}-\frac{45\!\cdots\!87}{25\!\cdots\!78}a^{14}+\frac{23\!\cdots\!32}{12\!\cdots\!39}a^{13}+\frac{72\!\cdots\!39}{12\!\cdots\!39}a^{12}-\frac{60\!\cdots\!32}{12\!\cdots\!39}a^{11}-\frac{12\!\cdots\!70}{12\!\cdots\!39}a^{10}+\frac{88\!\cdots\!97}{12\!\cdots\!39}a^{9}+\frac{24\!\cdots\!71}{25\!\cdots\!78}a^{8}-\frac{72\!\cdots\!48}{12\!\cdots\!39}a^{7}-\frac{64\!\cdots\!93}{12\!\cdots\!39}a^{6}+\frac{30\!\cdots\!12}{12\!\cdots\!39}a^{5}+\frac{16\!\cdots\!15}{12\!\cdots\!39}a^{4}-\frac{56\!\cdots\!14}{12\!\cdots\!39}a^{3}-\frac{36\!\cdots\!07}{25\!\cdots\!78}a^{2}+\frac{31\!\cdots\!11}{12\!\cdots\!39}a+\frac{63\!\cdots\!45}{12\!\cdots\!39}$, $\frac{15\!\cdots\!83}{25\!\cdots\!78}a^{20}-\frac{14\!\cdots\!24}{12\!\cdots\!39}a^{19}-\frac{26\!\cdots\!38}{12\!\cdots\!39}a^{18}+\frac{27\!\cdots\!70}{12\!\cdots\!39}a^{17}+\frac{32\!\cdots\!66}{12\!\cdots\!39}a^{16}-\frac{14\!\cdots\!49}{12\!\cdots\!39}a^{15}-\frac{37\!\cdots\!73}{25\!\cdots\!78}a^{14}+\frac{18\!\cdots\!50}{12\!\cdots\!39}a^{13}+\frac{53\!\cdots\!13}{12\!\cdots\!39}a^{12}+\frac{10\!\cdots\!86}{12\!\cdots\!39}a^{11}-\frac{75\!\cdots\!31}{12\!\cdots\!39}a^{10}-\frac{42\!\cdots\!76}{12\!\cdots\!39}a^{9}+\frac{85\!\cdots\!15}{25\!\cdots\!78}a^{8}+\frac{69\!\cdots\!40}{12\!\cdots\!39}a^{7}+\frac{65\!\cdots\!15}{12\!\cdots\!39}a^{6}-\frac{55\!\cdots\!87}{12\!\cdots\!39}a^{5}-\frac{16\!\cdots\!20}{12\!\cdots\!39}a^{4}+\frac{20\!\cdots\!82}{12\!\cdots\!39}a^{3}+\frac{10\!\cdots\!39}{25\!\cdots\!78}a^{2}-\frac{26\!\cdots\!36}{12\!\cdots\!39}a-\frac{43\!\cdots\!25}{12\!\cdots\!39}$, $\frac{36\!\cdots\!41}{12\!\cdots\!39}a^{20}-\frac{90\!\cdots\!74}{12\!\cdots\!39}a^{19}-\frac{12\!\cdots\!27}{12\!\cdots\!39}a^{18}+\frac{21\!\cdots\!72}{12\!\cdots\!39}a^{17}+\frac{15\!\cdots\!47}{12\!\cdots\!39}a^{16}-\frac{19\!\cdots\!14}{12\!\cdots\!39}a^{15}-\frac{91\!\cdots\!94}{12\!\cdots\!39}a^{14}+\frac{89\!\cdots\!35}{12\!\cdots\!39}a^{13}+\frac{29\!\cdots\!06}{12\!\cdots\!39}a^{12}-\frac{23\!\cdots\!30}{12\!\cdots\!39}a^{11}-\frac{52\!\cdots\!29}{12\!\cdots\!39}a^{10}+\frac{35\!\cdots\!86}{12\!\cdots\!39}a^{9}+\frac{52\!\cdots\!83}{12\!\cdots\!39}a^{8}-\frac{29\!\cdots\!78}{12\!\cdots\!39}a^{7}-\frac{28\!\cdots\!70}{12\!\cdots\!39}a^{6}+\frac{12\!\cdots\!42}{12\!\cdots\!39}a^{5}+\frac{77\!\cdots\!46}{12\!\cdots\!39}a^{4}-\frac{26\!\cdots\!10}{12\!\cdots\!39}a^{3}-\frac{97\!\cdots\!56}{12\!\cdots\!39}a^{2}+\frac{18\!\cdots\!58}{12\!\cdots\!39}a+\frac{39\!\cdots\!47}{12\!\cdots\!39}$, $\frac{27\!\cdots\!65}{25\!\cdots\!78}a^{20}-\frac{35\!\cdots\!93}{12\!\cdots\!39}a^{19}-\frac{45\!\cdots\!77}{12\!\cdots\!39}a^{18}+\frac{85\!\cdots\!11}{12\!\cdots\!39}a^{17}+\frac{56\!\cdots\!17}{12\!\cdots\!39}a^{16}-\frac{77\!\cdots\!05}{12\!\cdots\!39}a^{15}-\frac{67\!\cdots\!43}{25\!\cdots\!78}a^{14}+\frac{35\!\cdots\!29}{12\!\cdots\!39}a^{13}+\frac{10\!\cdots\!17}{12\!\cdots\!39}a^{12}-\frac{93\!\cdots\!86}{12\!\cdots\!39}a^{11}-\frac{18\!\cdots\!04}{12\!\cdots\!39}a^{10}+\frac{13\!\cdots\!32}{12\!\cdots\!39}a^{9}+\frac{36\!\cdots\!07}{25\!\cdots\!78}a^{8}-\frac{11\!\cdots\!24}{12\!\cdots\!39}a^{7}-\frac{95\!\cdots\!31}{12\!\cdots\!39}a^{6}+\frac{49\!\cdots\!88}{12\!\cdots\!39}a^{5}+\frac{24\!\cdots\!71}{12\!\cdots\!39}a^{4}-\frac{92\!\cdots\!21}{12\!\cdots\!39}a^{3}-\frac{55\!\cdots\!83}{25\!\cdots\!78}a^{2}+\frac{55\!\cdots\!16}{12\!\cdots\!39}a+\frac{10\!\cdots\!59}{12\!\cdots\!39}$, $\frac{56\!\cdots\!48}{12\!\cdots\!39}a^{20}-\frac{27\!\cdots\!55}{12\!\cdots\!39}a^{19}-\frac{14\!\cdots\!08}{12\!\cdots\!39}a^{18}+\frac{76\!\cdots\!58}{12\!\cdots\!39}a^{17}+\frac{13\!\cdots\!61}{12\!\cdots\!39}a^{16}-\frac{81\!\cdots\!42}{12\!\cdots\!39}a^{15}-\frac{42\!\cdots\!90}{12\!\cdots\!39}a^{14}+\frac{42\!\cdots\!82}{12\!\cdots\!39}a^{13}-\frac{30\!\cdots\!34}{12\!\cdots\!39}a^{12}-\frac{11\!\cdots\!91}{12\!\cdots\!39}a^{11}+\frac{53\!\cdots\!12}{12\!\cdots\!39}a^{10}+\frac{16\!\cdots\!29}{12\!\cdots\!39}a^{9}-\frac{13\!\cdots\!46}{12\!\cdots\!39}a^{8}-\frac{98\!\cdots\!01}{12\!\cdots\!39}a^{7}+\frac{14\!\cdots\!85}{12\!\cdots\!39}a^{6}+\frac{68\!\cdots\!17}{12\!\cdots\!39}a^{5}-\frac{74\!\cdots\!87}{12\!\cdots\!39}a^{4}+\frac{13\!\cdots\!50}{12\!\cdots\!39}a^{3}+\frac{15\!\cdots\!92}{12\!\cdots\!39}a^{2}-\frac{24\!\cdots\!04}{12\!\cdots\!39}a-\frac{76\!\cdots\!07}{12\!\cdots\!39}$, $\frac{67\!\cdots\!87}{25\!\cdots\!78}a^{20}-\frac{86\!\cdots\!78}{12\!\cdots\!39}a^{19}-\frac{11\!\cdots\!23}{12\!\cdots\!39}a^{18}+\frac{20\!\cdots\!59}{12\!\cdots\!39}a^{17}+\frac{13\!\cdots\!34}{12\!\cdots\!39}a^{16}-\frac{18\!\cdots\!27}{12\!\cdots\!39}a^{15}-\frac{16\!\cdots\!73}{25\!\cdots\!78}a^{14}+\frac{88\!\cdots\!95}{12\!\cdots\!39}a^{13}+\frac{26\!\cdots\!58}{12\!\cdots\!39}a^{12}-\frac{23\!\cdots\!75}{12\!\cdots\!39}a^{11}-\frac{45\!\cdots\!62}{12\!\cdots\!39}a^{10}+\frac{34\!\cdots\!89}{12\!\cdots\!39}a^{9}+\frac{87\!\cdots\!67}{25\!\cdots\!78}a^{8}-\frac{28\!\cdots\!20}{12\!\cdots\!39}a^{7}-\frac{22\!\cdots\!57}{12\!\cdots\!39}a^{6}+\frac{11\!\cdots\!21}{12\!\cdots\!39}a^{5}+\frac{57\!\cdots\!05}{12\!\cdots\!39}a^{4}-\frac{22\!\cdots\!51}{12\!\cdots\!39}a^{3}-\frac{12\!\cdots\!93}{25\!\cdots\!78}a^{2}+\frac{13\!\cdots\!91}{12\!\cdots\!39}a+\frac{25\!\cdots\!63}{12\!\cdots\!39}$, $\frac{15\!\cdots\!40}{12\!\cdots\!39}a^{20}-\frac{38\!\cdots\!16}{12\!\cdots\!39}a^{19}-\frac{49\!\cdots\!83}{12\!\cdots\!39}a^{18}+\frac{91\!\cdots\!99}{12\!\cdots\!39}a^{17}+\frac{62\!\cdots\!59}{12\!\cdots\!39}a^{16}-\frac{82\!\cdots\!04}{12\!\cdots\!39}a^{15}-\frac{37\!\cdots\!34}{12\!\cdots\!39}a^{14}+\frac{37\!\cdots\!84}{12\!\cdots\!39}a^{13}+\frac{11\!\cdots\!15}{12\!\cdots\!39}a^{12}-\frac{98\!\cdots\!82}{12\!\cdots\!39}a^{11}-\frac{20\!\cdots\!02}{12\!\cdots\!39}a^{10}+\frac{14\!\cdots\!05}{12\!\cdots\!39}a^{9}+\frac{20\!\cdots\!82}{12\!\cdots\!39}a^{8}-\frac{11\!\cdots\!18}{12\!\cdots\!39}a^{7}-\frac{10\!\cdots\!58}{12\!\cdots\!39}a^{6}+\frac{50\!\cdots\!20}{12\!\cdots\!39}a^{5}+\frac{27\!\cdots\!79}{12\!\cdots\!39}a^{4}-\frac{94\!\cdots\!85}{12\!\cdots\!39}a^{3}-\frac{30\!\cdots\!73}{12\!\cdots\!39}a^{2}+\frac{56\!\cdots\!78}{12\!\cdots\!39}a+\frac{11\!\cdots\!67}{12\!\cdots\!39}$, $\frac{24\!\cdots\!39}{25\!\cdots\!78}a^{20}-\frac{23\!\cdots\!58}{12\!\cdots\!39}a^{19}-\frac{40\!\cdots\!46}{12\!\cdots\!39}a^{18}+\frac{51\!\cdots\!96}{12\!\cdots\!39}a^{17}+\frac{52\!\cdots\!56}{12\!\cdots\!39}a^{16}-\frac{37\!\cdots\!42}{12\!\cdots\!39}a^{15}-\frac{64\!\cdots\!73}{25\!\cdots\!78}a^{14}+\frac{13\!\cdots\!89}{12\!\cdots\!39}a^{13}+\frac{10\!\cdots\!50}{12\!\cdots\!39}a^{12}-\frac{27\!\cdots\!22}{12\!\cdots\!39}a^{11}-\frac{18\!\cdots\!05}{12\!\cdots\!39}a^{10}+\frac{30\!\cdots\!33}{12\!\cdots\!39}a^{9}+\frac{38\!\cdots\!99}{25\!\cdots\!78}a^{8}-\frac{16\!\cdots\!64}{12\!\cdots\!39}a^{7}-\frac{10\!\cdots\!67}{12\!\cdots\!39}a^{6}+\frac{19\!\cdots\!75}{12\!\cdots\!39}a^{5}+\frac{29\!\cdots\!63}{12\!\cdots\!39}a^{4}+\frac{13\!\cdots\!64}{12\!\cdots\!39}a^{3}-\frac{70\!\cdots\!57}{25\!\cdots\!78}a^{2}-\frac{29\!\cdots\!55}{12\!\cdots\!39}a+\frac{10\!\cdots\!63}{12\!\cdots\!39}$, $\frac{38\!\cdots\!64}{12\!\cdots\!39}a^{20}-\frac{94\!\cdots\!77}{12\!\cdots\!39}a^{19}-\frac{12\!\cdots\!46}{12\!\cdots\!39}a^{18}+\frac{22\!\cdots\!81}{12\!\cdots\!39}a^{17}+\frac{15\!\cdots\!68}{12\!\cdots\!39}a^{16}-\frac{20\!\cdots\!82}{12\!\cdots\!39}a^{15}-\frac{96\!\cdots\!06}{12\!\cdots\!39}a^{14}+\frac{94\!\cdots\!46}{12\!\cdots\!39}a^{13}+\frac{31\!\cdots\!17}{12\!\cdots\!39}a^{12}-\frac{24\!\cdots\!60}{12\!\cdots\!39}a^{11}-\frac{56\!\cdots\!77}{12\!\cdots\!39}a^{10}+\frac{37\!\cdots\!93}{12\!\cdots\!39}a^{9}+\frac{57\!\cdots\!22}{12\!\cdots\!39}a^{8}-\frac{31\!\cdots\!38}{12\!\cdots\!39}a^{7}-\frac{32\!\cdots\!63}{12\!\cdots\!39}a^{6}+\frac{13\!\cdots\!95}{12\!\cdots\!39}a^{5}+\frac{94\!\cdots\!03}{12\!\cdots\!39}a^{4}-\frac{27\!\cdots\!44}{12\!\cdots\!39}a^{3}-\frac{12\!\cdots\!15}{12\!\cdots\!39}a^{2}+\frac{19\!\cdots\!33}{12\!\cdots\!39}a+\frac{44\!\cdots\!27}{12\!\cdots\!39}$
| 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: | \( 8566215956620 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 8566215956620 \cdot 1}{2\cdot\sqrt{114304429232803182734215052479732645888}}\cr\approx \mathstrut & 0.840151104106996 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4)
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 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4, 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 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4);
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 - 2*x^20 - 34*x^19 + 44*x^18 + 440*x^17 - 336*x^16 - 2735*x^15 + 1258*x^14 + 9114*x^13 - 2548*x^12 - 17108*x^11 + 2708*x^10 + 18395*x^9 - 1186*x^8 - 11144*x^7 - 160*x^6 + 3584*x^5 + 284*x^4 - 539*x^3 - 66*x^2 + 28*x + 4);
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
$S_3\times \GL(3,2)$ (as 21T27):
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 1008 |
| The 18 conjugacy class representatives for $S_3\times \GL(3,2)$ |
| Character table for $S_3\times \GL(3,2)$ |
Intermediate fields
| 3.3.148.1, 7.7.670188544.1 |
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 24 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
| Minimal sibling: | 21.21.114304429232803182734215052479732645888.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $21$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | $21$ | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | R | $21$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{7}$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.10.2 | $x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2$ | $6$ | $1$ | $10$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 2.12.20.37 | $x^{12} + 10 x^{11} + 51 x^{10} + 176 x^{9} + 450 x^{8} + 870 x^{7} + 1299 x^{6} + 1516 x^{5} + 1250 x^{4} + 542 x^{3} + 67 x^{2} - 56 x + 7$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(809\)
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |