Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5)
gp: K = bnfinit(y^21 - 2*y^19 - 7*y^18 - 5*y^17 + 11*y^16 + 17*y^15 + 34*y^14 - 32*y^13 - 19*y^12 - 120*y^11 - 65*y^10 - 276*y^9 + 43*y^8 + 539*y^7 + 983*y^6 + 606*y^5 + 280*y^4 + 121*y^3 + 40*y^2 - 13*y - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5)
\( x^{21} - 2 x^{19} - 7 x^{18} - 5 x^{17} + 11 x^{16} + 17 x^{15} + 34 x^{14} - 32 x^{13} - 19 x^{12} + \cdots - 5 \)
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: | $[5, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(27984468103350046268523020288\)
\(\medspace = 2^{27}\cdot 181^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.63\) | 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: | $2^{3/2}181^{1/2}\approx 38.05259518088089$ | ||
| Ramified primes: |
\(2\), \(181\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{362}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{17}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{41\!\cdots\!64}a^{20}-\frac{43\!\cdots\!17}{41\!\cdots\!64}a^{19}+\frac{49\!\cdots\!41}{41\!\cdots\!64}a^{18}-\frac{81\!\cdots\!89}{41\!\cdots\!64}a^{17}-\frac{18\!\cdots\!91}{20\!\cdots\!82}a^{16}-\frac{19\!\cdots\!12}{10\!\cdots\!41}a^{15}+\frac{86\!\cdots\!15}{41\!\cdots\!64}a^{14}+\frac{24\!\cdots\!95}{41\!\cdots\!64}a^{13}-\frac{37\!\cdots\!41}{41\!\cdots\!64}a^{12}-\frac{18\!\cdots\!27}{41\!\cdots\!64}a^{11}-\frac{37\!\cdots\!47}{41\!\cdots\!64}a^{10}+\frac{11\!\cdots\!73}{41\!\cdots\!64}a^{9}+\frac{13\!\cdots\!51}{41\!\cdots\!64}a^{8}+\frac{43\!\cdots\!11}{41\!\cdots\!64}a^{7}-\frac{50\!\cdots\!27}{10\!\cdots\!41}a^{6}+\frac{13\!\cdots\!08}{10\!\cdots\!41}a^{5}+\frac{17\!\cdots\!67}{10\!\cdots\!41}a^{4}+\frac{21\!\cdots\!77}{10\!\cdots\!41}a^{3}+\frac{37\!\cdots\!13}{41\!\cdots\!64}a^{2}+\frac{73\!\cdots\!09}{41\!\cdots\!64}a+\frac{49\!\cdots\!58}{10\!\cdots\!41}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $12$ | 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: |
$\frac{15\!\cdots\!79}{20\!\cdots\!82}a^{20}-\frac{34\!\cdots\!79}{20\!\cdots\!82}a^{19}-\frac{61\!\cdots\!13}{41\!\cdots\!64}a^{18}-\frac{10\!\cdots\!43}{20\!\cdots\!82}a^{17}-\frac{11\!\cdots\!05}{41\!\cdots\!64}a^{16}+\frac{92\!\cdots\!37}{10\!\cdots\!41}a^{15}+\frac{22\!\cdots\!57}{20\!\cdots\!82}a^{14}+\frac{48\!\cdots\!29}{20\!\cdots\!82}a^{13}-\frac{12\!\cdots\!07}{41\!\cdots\!64}a^{12}-\frac{16\!\cdots\!23}{20\!\cdots\!82}a^{11}-\frac{37\!\cdots\!27}{41\!\cdots\!64}a^{10}-\frac{30\!\cdots\!28}{10\!\cdots\!41}a^{9}-\frac{83\!\cdots\!19}{41\!\cdots\!64}a^{8}+\frac{82\!\cdots\!22}{10\!\cdots\!41}a^{7}+\frac{16\!\cdots\!29}{41\!\cdots\!64}a^{6}+\frac{13\!\cdots\!27}{20\!\cdots\!82}a^{5}+\frac{64\!\cdots\!63}{20\!\cdots\!82}a^{4}+\frac{15\!\cdots\!03}{10\!\cdots\!41}a^{3}+\frac{42\!\cdots\!23}{10\!\cdots\!41}a^{2}-\frac{28\!\cdots\!26}{10\!\cdots\!41}a-\frac{87\!\cdots\!23}{41\!\cdots\!64}$, $\frac{19\!\cdots\!79}{20\!\cdots\!82}a^{20}-\frac{20\!\cdots\!45}{41\!\cdots\!64}a^{19}-\frac{17\!\cdots\!85}{10\!\cdots\!41}a^{18}-\frac{23\!\cdots\!99}{41\!\cdots\!64}a^{17}-\frac{22\!\cdots\!49}{10\!\cdots\!41}a^{16}+\frac{11\!\cdots\!26}{10\!\cdots\!41}a^{15}+\frac{10\!\cdots\!42}{10\!\cdots\!41}a^{14}+\frac{11\!\cdots\!25}{41\!\cdots\!64}a^{13}-\frac{85\!\cdots\!69}{20\!\cdots\!82}a^{12}+\frac{11\!\cdots\!45}{41\!\cdots\!64}a^{11}-\frac{11\!\cdots\!58}{10\!\cdots\!41}a^{10}-\frac{69\!\cdots\!79}{41\!\cdots\!64}a^{9}-\frac{26\!\cdots\!83}{10\!\cdots\!41}a^{8}+\frac{65\!\cdots\!67}{41\!\cdots\!64}a^{7}+\frac{91\!\cdots\!57}{20\!\cdots\!82}a^{6}+\frac{14\!\cdots\!87}{20\!\cdots\!82}a^{5}+\frac{57\!\cdots\!55}{20\!\cdots\!82}a^{4}+\frac{57\!\cdots\!39}{20\!\cdots\!82}a^{3}-\frac{13\!\cdots\!33}{10\!\cdots\!41}a^{2}-\frac{13\!\cdots\!77}{41\!\cdots\!64}a-\frac{75\!\cdots\!67}{20\!\cdots\!82}$, $\frac{64\!\cdots\!84}{10\!\cdots\!41}a^{20}-\frac{38\!\cdots\!66}{10\!\cdots\!41}a^{19}-\frac{44\!\cdots\!39}{41\!\cdots\!64}a^{18}-\frac{37\!\cdots\!73}{10\!\cdots\!41}a^{17}-\frac{36\!\cdots\!97}{41\!\cdots\!64}a^{16}+\frac{79\!\cdots\!25}{10\!\cdots\!41}a^{15}+\frac{12\!\cdots\!33}{20\!\cdots\!82}a^{14}+\frac{17\!\cdots\!39}{10\!\cdots\!41}a^{13}-\frac{12\!\cdots\!17}{41\!\cdots\!64}a^{12}+\frac{52\!\cdots\!63}{10\!\cdots\!41}a^{11}-\frac{31\!\cdots\!77}{41\!\cdots\!64}a^{10}+\frac{41\!\cdots\!03}{20\!\cdots\!82}a^{9}-\frac{68\!\cdots\!45}{41\!\cdots\!64}a^{8}+\frac{25\!\cdots\!83}{20\!\cdots\!82}a^{7}+\frac{11\!\cdots\!49}{41\!\cdots\!64}a^{6}+\frac{89\!\cdots\!23}{20\!\cdots\!82}a^{5}+\frac{23\!\cdots\!51}{20\!\cdots\!82}a^{4}+\frac{85\!\cdots\!92}{10\!\cdots\!41}a^{3}+\frac{46\!\cdots\!55}{20\!\cdots\!82}a^{2}-\frac{13\!\cdots\!59}{20\!\cdots\!82}a-\frac{53\!\cdots\!01}{41\!\cdots\!64}$, $\frac{34\!\cdots\!67}{41\!\cdots\!64}a^{20}+\frac{70\!\cdots\!71}{41\!\cdots\!64}a^{19}-\frac{23\!\cdots\!73}{20\!\cdots\!82}a^{18}-\frac{37\!\cdots\!69}{41\!\cdots\!64}a^{17}-\frac{71\!\cdots\!61}{41\!\cdots\!64}a^{16}-\frac{54\!\cdots\!83}{20\!\cdots\!82}a^{15}+\frac{12\!\cdots\!23}{41\!\cdots\!64}a^{14}+\frac{26\!\cdots\!27}{41\!\cdots\!64}a^{13}+\frac{82\!\cdots\!17}{20\!\cdots\!82}a^{12}-\frac{23\!\cdots\!99}{41\!\cdots\!64}a^{11}-\frac{15\!\cdots\!95}{10\!\cdots\!41}a^{10}-\frac{11\!\cdots\!37}{41\!\cdots\!64}a^{9}-\frac{77\!\cdots\!57}{20\!\cdots\!82}a^{8}-\frac{19\!\cdots\!03}{41\!\cdots\!64}a^{7}+\frac{17\!\cdots\!67}{41\!\cdots\!64}a^{6}+\frac{35\!\cdots\!23}{20\!\cdots\!82}a^{5}+\frac{51\!\cdots\!59}{20\!\cdots\!82}a^{4}+\frac{17\!\cdots\!29}{10\!\cdots\!41}a^{3}+\frac{33\!\cdots\!17}{41\!\cdots\!64}a^{2}+\frac{10\!\cdots\!93}{41\!\cdots\!64}a+\frac{62\!\cdots\!13}{41\!\cdots\!64}$, $\frac{24\!\cdots\!14}{10\!\cdots\!41}a^{20}-\frac{25\!\cdots\!63}{10\!\cdots\!41}a^{19}-\frac{10\!\cdots\!33}{20\!\cdots\!82}a^{18}-\frac{11\!\cdots\!19}{10\!\cdots\!41}a^{17}+\frac{61\!\cdots\!45}{10\!\cdots\!41}a^{16}+\frac{40\!\cdots\!12}{10\!\cdots\!41}a^{15}+\frac{13\!\cdots\!29}{10\!\cdots\!41}a^{14}+\frac{37\!\cdots\!20}{10\!\cdots\!41}a^{13}-\frac{16\!\cdots\!10}{10\!\cdots\!41}a^{12}+\frac{31\!\cdots\!58}{10\!\cdots\!41}a^{11}-\frac{23\!\cdots\!93}{10\!\cdots\!41}a^{10}+\frac{14\!\cdots\!21}{10\!\cdots\!41}a^{9}-\frac{49\!\cdots\!33}{10\!\cdots\!41}a^{8}+\frac{81\!\cdots\!78}{10\!\cdots\!41}a^{7}+\frac{12\!\cdots\!12}{10\!\cdots\!41}a^{6}+\frac{97\!\cdots\!36}{10\!\cdots\!41}a^{5}-\frac{22\!\cdots\!03}{20\!\cdots\!82}a^{4}-\frac{94\!\cdots\!02}{10\!\cdots\!41}a^{3}-\frac{38\!\cdots\!79}{10\!\cdots\!41}a^{2}-\frac{22\!\cdots\!36}{10\!\cdots\!41}a-\frac{15\!\cdots\!63}{10\!\cdots\!41}$, $\frac{56\!\cdots\!77}{41\!\cdots\!64}a^{20}-\frac{23\!\cdots\!94}{10\!\cdots\!41}a^{19}-\frac{11\!\cdots\!31}{41\!\cdots\!64}a^{18}-\frac{18\!\cdots\!91}{20\!\cdots\!82}a^{17}-\frac{11\!\cdots\!15}{20\!\cdots\!82}a^{16}+\frac{33\!\cdots\!87}{20\!\cdots\!82}a^{15}+\frac{85\!\cdots\!71}{41\!\cdots\!64}a^{14}+\frac{89\!\cdots\!73}{20\!\cdots\!82}a^{13}-\frac{21\!\cdots\!83}{41\!\cdots\!64}a^{12}-\frac{18\!\cdots\!83}{10\!\cdots\!41}a^{11}-\frac{66\!\cdots\!47}{41\!\cdots\!64}a^{10}-\frac{64\!\cdots\!98}{10\!\cdots\!41}a^{9}-\frac{15\!\cdots\!77}{41\!\cdots\!64}a^{8}+\frac{12\!\cdots\!09}{10\!\cdots\!41}a^{7}+\frac{74\!\cdots\!41}{10\!\cdots\!41}a^{6}+\frac{25\!\cdots\!55}{20\!\cdots\!82}a^{5}+\frac{13\!\cdots\!45}{20\!\cdots\!82}a^{4}+\frac{55\!\cdots\!47}{20\!\cdots\!82}a^{3}+\frac{45\!\cdots\!85}{41\!\cdots\!64}a^{2}+\frac{28\!\cdots\!87}{10\!\cdots\!41}a-\frac{43\!\cdots\!31}{20\!\cdots\!82}$, $\frac{74\!\cdots\!01}{41\!\cdots\!64}a^{20}+\frac{41\!\cdots\!67}{20\!\cdots\!82}a^{19}-\frac{23\!\cdots\!49}{41\!\cdots\!64}a^{18}-\frac{31\!\cdots\!41}{20\!\cdots\!82}a^{17}-\frac{39\!\cdots\!03}{20\!\cdots\!82}a^{16}+\frac{43\!\cdots\!13}{20\!\cdots\!82}a^{15}+\frac{22\!\cdots\!27}{41\!\cdots\!64}a^{14}+\frac{70\!\cdots\!96}{10\!\cdots\!41}a^{13}-\frac{31\!\cdots\!61}{41\!\cdots\!64}a^{12}-\frac{30\!\cdots\!73}{20\!\cdots\!82}a^{11}-\frac{61\!\cdots\!97}{41\!\cdots\!64}a^{10}-\frac{76\!\cdots\!39}{20\!\cdots\!82}a^{9}-\frac{16\!\cdots\!35}{41\!\cdots\!64}a^{8}-\frac{10\!\cdots\!85}{20\!\cdots\!82}a^{7}+\frac{31\!\cdots\!63}{20\!\cdots\!82}a^{6}+\frac{50\!\cdots\!81}{20\!\cdots\!82}a^{5}+\frac{43\!\cdots\!31}{20\!\cdots\!82}a^{4}+\frac{51\!\cdots\!86}{10\!\cdots\!41}a^{3}+\frac{28\!\cdots\!63}{41\!\cdots\!64}a^{2}+\frac{87\!\cdots\!81}{20\!\cdots\!82}a+\frac{34\!\cdots\!77}{10\!\cdots\!41}$, $\frac{47\!\cdots\!59}{41\!\cdots\!64}a^{20}-\frac{88\!\cdots\!27}{20\!\cdots\!82}a^{19}-\frac{45\!\cdots\!29}{20\!\cdots\!82}a^{18}-\frac{73\!\cdots\!53}{10\!\cdots\!41}a^{17}-\frac{11\!\cdots\!45}{41\!\cdots\!64}a^{16}+\frac{14\!\cdots\!61}{10\!\cdots\!41}a^{15}+\frac{59\!\cdots\!93}{41\!\cdots\!64}a^{14}+\frac{33\!\cdots\!73}{10\!\cdots\!41}a^{13}-\frac{10\!\cdots\!03}{20\!\cdots\!82}a^{12}-\frac{12\!\cdots\!51}{20\!\cdots\!82}a^{11}-\frac{27\!\cdots\!55}{20\!\cdots\!82}a^{10}-\frac{26\!\cdots\!22}{10\!\cdots\!41}a^{9}-\frac{30\!\cdots\!22}{10\!\cdots\!41}a^{8}+\frac{16\!\cdots\!64}{10\!\cdots\!41}a^{7}+\frac{24\!\cdots\!37}{41\!\cdots\!64}a^{6}+\frac{92\!\cdots\!94}{10\!\cdots\!41}a^{5}+\frac{65\!\cdots\!75}{20\!\cdots\!82}a^{4}+\frac{25\!\cdots\!15}{20\!\cdots\!82}a^{3}+\frac{19\!\cdots\!15}{41\!\cdots\!64}a^{2}+\frac{10\!\cdots\!59}{20\!\cdots\!82}a-\frac{11\!\cdots\!61}{41\!\cdots\!64}$, $\frac{76\!\cdots\!73}{41\!\cdots\!64}a^{20}-\frac{11\!\cdots\!21}{20\!\cdots\!82}a^{19}-\frac{36\!\cdots\!21}{10\!\cdots\!41}a^{18}-\frac{72\!\cdots\!32}{10\!\cdots\!41}a^{17}+\frac{11\!\cdots\!17}{41\!\cdots\!64}a^{16}+\frac{42\!\cdots\!23}{10\!\cdots\!41}a^{15}-\frac{17\!\cdots\!27}{41\!\cdots\!64}a^{14}-\frac{32\!\cdots\!27}{10\!\cdots\!41}a^{13}-\frac{21\!\cdots\!08}{10\!\cdots\!41}a^{12}+\frac{35\!\cdots\!27}{20\!\cdots\!82}a^{11}-\frac{66\!\cdots\!71}{10\!\cdots\!41}a^{10}+\frac{43\!\cdots\!93}{10\!\cdots\!41}a^{9}-\frac{44\!\cdots\!73}{20\!\cdots\!82}a^{8}+\frac{13\!\cdots\!53}{10\!\cdots\!41}a^{7}+\frac{26\!\cdots\!61}{41\!\cdots\!64}a^{6}-\frac{16\!\cdots\!60}{10\!\cdots\!41}a^{5}-\frac{76\!\cdots\!87}{20\!\cdots\!82}a^{4}-\frac{27\!\cdots\!39}{20\!\cdots\!82}a^{3}+\frac{26\!\cdots\!95}{41\!\cdots\!64}a^{2}+\frac{32\!\cdots\!47}{20\!\cdots\!82}a-\frac{42\!\cdots\!51}{41\!\cdots\!64}$, $\frac{10\!\cdots\!49}{20\!\cdots\!82}a^{20}-\frac{20\!\cdots\!49}{10\!\cdots\!41}a^{19}-\frac{10\!\cdots\!86}{10\!\cdots\!41}a^{18}-\frac{35\!\cdots\!90}{10\!\cdots\!41}a^{17}-\frac{23\!\cdots\!43}{10\!\cdots\!41}a^{16}+\frac{58\!\cdots\!66}{10\!\cdots\!41}a^{15}+\frac{17\!\cdots\!03}{20\!\cdots\!82}a^{14}+\frac{16\!\cdots\!64}{10\!\cdots\!41}a^{13}-\frac{17\!\cdots\!37}{10\!\cdots\!41}a^{12}-\frac{90\!\cdots\!90}{10\!\cdots\!41}a^{11}-\frac{61\!\cdots\!73}{10\!\cdots\!41}a^{10}-\frac{29\!\cdots\!45}{10\!\cdots\!41}a^{9}-\frac{14\!\cdots\!03}{10\!\cdots\!41}a^{8}+\frac{32\!\cdots\!77}{10\!\cdots\!41}a^{7}+\frac{55\!\cdots\!53}{20\!\cdots\!82}a^{6}+\frac{50\!\cdots\!06}{10\!\cdots\!41}a^{5}+\frac{26\!\cdots\!83}{10\!\cdots\!41}a^{4}+\frac{13\!\cdots\!34}{10\!\cdots\!41}a^{3}+\frac{69\!\cdots\!02}{10\!\cdots\!41}a^{2}+\frac{15\!\cdots\!57}{10\!\cdots\!41}a-\frac{26\!\cdots\!43}{20\!\cdots\!82}$, $\frac{64\!\cdots\!64}{10\!\cdots\!41}a^{20}-\frac{36\!\cdots\!07}{20\!\cdots\!82}a^{19}-\frac{57\!\cdots\!79}{41\!\cdots\!64}a^{18}-\frac{81\!\cdots\!33}{20\!\cdots\!82}a^{17}-\frac{65\!\cdots\!03}{41\!\cdots\!64}a^{16}+\frac{17\!\cdots\!31}{20\!\cdots\!82}a^{15}+\frac{93\!\cdots\!37}{10\!\cdots\!41}a^{14}+\frac{33\!\cdots\!21}{20\!\cdots\!82}a^{13}-\frac{11\!\cdots\!97}{41\!\cdots\!64}a^{12}-\frac{21\!\cdots\!09}{20\!\cdots\!82}a^{11}-\frac{26\!\cdots\!21}{41\!\cdots\!64}a^{10}-\frac{18\!\cdots\!18}{10\!\cdots\!41}a^{9}-\frac{59\!\cdots\!25}{41\!\cdots\!64}a^{8}+\frac{80\!\cdots\!69}{10\!\cdots\!41}a^{7}+\frac{14\!\cdots\!33}{41\!\cdots\!64}a^{6}+\frac{10\!\cdots\!23}{20\!\cdots\!82}a^{5}+\frac{26\!\cdots\!17}{20\!\cdots\!82}a^{4}-\frac{45\!\cdots\!91}{10\!\cdots\!41}a^{3}-\frac{87\!\cdots\!66}{10\!\cdots\!41}a^{2}+\frac{16\!\cdots\!69}{20\!\cdots\!82}a-\frac{35\!\cdots\!19}{41\!\cdots\!64}$, $\frac{16\!\cdots\!25}{41\!\cdots\!64}a^{20}-\frac{11\!\cdots\!31}{41\!\cdots\!64}a^{19}-\frac{23\!\cdots\!41}{41\!\cdots\!64}a^{18}-\frac{94\!\cdots\!31}{41\!\cdots\!64}a^{17}-\frac{44\!\cdots\!37}{10\!\cdots\!41}a^{16}+\frac{90\!\cdots\!27}{20\!\cdots\!82}a^{15}+\frac{12\!\cdots\!57}{41\!\cdots\!64}a^{14}+\frac{46\!\cdots\!03}{41\!\cdots\!64}a^{13}-\frac{80\!\cdots\!41}{41\!\cdots\!64}a^{12}+\frac{33\!\cdots\!89}{41\!\cdots\!64}a^{11}-\frac{21\!\cdots\!55}{41\!\cdots\!64}a^{10}+\frac{24\!\cdots\!89}{41\!\cdots\!64}a^{9}-\frac{48\!\cdots\!73}{41\!\cdots\!64}a^{8}+\frac{36\!\cdots\!19}{41\!\cdots\!64}a^{7}+\frac{15\!\cdots\!08}{10\!\cdots\!41}a^{6}+\frac{55\!\cdots\!01}{20\!\cdots\!82}a^{5}+\frac{15\!\cdots\!81}{20\!\cdots\!82}a^{4}+\frac{25\!\cdots\!13}{20\!\cdots\!82}a^{3}+\frac{25\!\cdots\!39}{41\!\cdots\!64}a^{2}+\frac{99\!\cdots\!39}{41\!\cdots\!64}a-\frac{70\!\cdots\!71}{20\!\cdots\!82}$
| 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: | \( 2296254.50823 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 2296254.50823 \cdot 1}{2\cdot\sqrt{27984468103350046268523020288}}\cr\approx \mathstrut & 0.533482801470 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5)
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^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5, 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^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5);
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^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5);
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
$\PGL(2,7)$ (as 21T20):
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 336 |
| The 9 conjugacy class representatives for $\PGL(2,7)$ |
| Character table for $\PGL(2,7)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 8 sibling: | 8.0.3036027392.1 |
| Degree 14 sibling: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 28 siblings: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 8.0.3036027392.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
|
\(181\)
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |