Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 10*y^18 + 6*y^17 - 74*y^16 + 171*y^15 - 136*y^14 - 322*y^13 + 921*y^12 - 653*y^11 - 699*y^10 + 1328*y^9 - 176*y^8 - 678*y^7 + 468*y^6 + 206*y^5 - 157*y^4 - 65*y^3 + 4*y^2 + 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*x + 1)
\( x^{20} - 5 x^{19} + 10 x^{18} + 6 x^{17} - 74 x^{16} + 171 x^{15} - 136 x^{14} - 322 x^{13} + 921 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(240663579466141115896800721\)
\(\medspace = 13^{10}\cdot 347^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.85\) | 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: | $13^{1/2}347^{1/2}\approx 67.16397844082793$ | ||
| Ramified primes: |
\(13\), \(347\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13}a^{18}+\frac{6}{13}a^{17}-\frac{3}{13}a^{16}+\frac{6}{13}a^{15}-\frac{5}{13}a^{14}+\frac{6}{13}a^{13}-\frac{3}{13}a^{11}+\frac{4}{13}a^{10}+\frac{5}{13}a^{9}+\frac{2}{13}a^{8}+\frac{6}{13}a^{7}+\frac{5}{13}a^{6}-\frac{5}{13}a^{5}+\frac{5}{13}a^{4}+\frac{6}{13}a^{3}-\frac{5}{13}a^{2}+\frac{4}{13}a+\frac{1}{13}$, $\frac{1}{31\!\cdots\!71}a^{19}+\frac{59\!\cdots\!08}{31\!\cdots\!71}a^{18}-\frac{14\!\cdots\!85}{31\!\cdots\!71}a^{17}+\frac{10\!\cdots\!56}{31\!\cdots\!71}a^{16}+\frac{61\!\cdots\!67}{31\!\cdots\!71}a^{15}-\frac{77\!\cdots\!74}{24\!\cdots\!67}a^{14}-\frac{39\!\cdots\!60}{31\!\cdots\!71}a^{13}+\frac{10\!\cdots\!78}{31\!\cdots\!71}a^{12}-\frac{14\!\cdots\!79}{31\!\cdots\!71}a^{11}+\frac{42\!\cdots\!97}{31\!\cdots\!71}a^{10}-\frac{47\!\cdots\!07}{31\!\cdots\!71}a^{9}-\frac{21\!\cdots\!06}{31\!\cdots\!71}a^{8}+\frac{81\!\cdots\!77}{31\!\cdots\!71}a^{7}-\frac{24\!\cdots\!31}{31\!\cdots\!71}a^{6}-\frac{87\!\cdots\!11}{31\!\cdots\!71}a^{5}+\frac{68\!\cdots\!45}{31\!\cdots\!71}a^{4}+\frac{76\!\cdots\!12}{31\!\cdots\!71}a^{3}+\frac{71\!\cdots\!48}{31\!\cdots\!71}a^{2}-\frac{41\!\cdots\!70}{31\!\cdots\!71}a-\frac{19\!\cdots\!94}{31\!\cdots\!71}$
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: |
$\frac{28\!\cdots\!90}{12\!\cdots\!03}a^{19}-\frac{14\!\cdots\!04}{12\!\cdots\!03}a^{18}+\frac{30\!\cdots\!07}{12\!\cdots\!03}a^{17}+\frac{15\!\cdots\!36}{12\!\cdots\!03}a^{16}-\frac{16\!\cdots\!92}{95\!\cdots\!31}a^{15}+\frac{52\!\cdots\!06}{12\!\cdots\!03}a^{14}-\frac{43\!\cdots\!08}{12\!\cdots\!03}a^{13}-\frac{94\!\cdots\!08}{12\!\cdots\!03}a^{12}+\frac{28\!\cdots\!52}{12\!\cdots\!03}a^{11}-\frac{21\!\cdots\!85}{12\!\cdots\!03}a^{10}-\frac{20\!\cdots\!69}{12\!\cdots\!03}a^{9}+\frac{45\!\cdots\!88}{12\!\cdots\!03}a^{8}-\frac{10\!\cdots\!63}{12\!\cdots\!03}a^{7}-\frac{23\!\cdots\!39}{12\!\cdots\!03}a^{6}+\frac{20\!\cdots\!37}{12\!\cdots\!03}a^{5}+\frac{38\!\cdots\!17}{12\!\cdots\!03}a^{4}-\frac{54\!\cdots\!69}{95\!\cdots\!31}a^{3}-\frac{22\!\cdots\!47}{12\!\cdots\!03}a^{2}+\frac{41\!\cdots\!51}{95\!\cdots\!31}a+\frac{10\!\cdots\!53}{12\!\cdots\!03}$, $\frac{74\!\cdots\!93}{12\!\cdots\!03}a^{19}-\frac{40\!\cdots\!85}{12\!\cdots\!03}a^{18}+\frac{89\!\cdots\!30}{12\!\cdots\!03}a^{17}+\frac{13\!\cdots\!45}{12\!\cdots\!03}a^{16}-\frac{43\!\cdots\!08}{95\!\cdots\!31}a^{15}+\frac{14\!\cdots\!19}{12\!\cdots\!03}a^{14}-\frac{15\!\cdots\!65}{12\!\cdots\!03}a^{13}-\frac{19\!\cdots\!74}{12\!\cdots\!03}a^{12}+\frac{76\!\cdots\!00}{12\!\cdots\!03}a^{11}-\frac{76\!\cdots\!93}{12\!\cdots\!03}a^{10}-\frac{27\!\cdots\!98}{12\!\cdots\!03}a^{9}+\frac{11\!\cdots\!86}{12\!\cdots\!03}a^{8}-\frac{53\!\cdots\!10}{12\!\cdots\!03}a^{7}-\frac{35\!\cdots\!37}{12\!\cdots\!03}a^{6}+\frac{49\!\cdots\!23}{12\!\cdots\!03}a^{5}-\frac{12\!\cdots\!67}{12\!\cdots\!03}a^{4}-\frac{10\!\cdots\!17}{95\!\cdots\!31}a^{3}-\frac{57\!\cdots\!23}{12\!\cdots\!03}a^{2}+\frac{75\!\cdots\!49}{95\!\cdots\!31}a+\frac{23\!\cdots\!67}{12\!\cdots\!03}$, $\frac{11\!\cdots\!15}{31\!\cdots\!71}a^{19}-\frac{64\!\cdots\!95}{31\!\cdots\!71}a^{18}+\frac{15\!\cdots\!11}{31\!\cdots\!71}a^{17}-\frac{21\!\cdots\!64}{31\!\cdots\!71}a^{16}-\frac{66\!\cdots\!92}{24\!\cdots\!67}a^{15}+\frac{25\!\cdots\!73}{31\!\cdots\!71}a^{14}-\frac{30\!\cdots\!77}{31\!\cdots\!71}a^{13}-\frac{21\!\cdots\!33}{31\!\cdots\!71}a^{12}+\frac{12\!\cdots\!91}{31\!\cdots\!71}a^{11}-\frac{15\!\cdots\!75}{31\!\cdots\!71}a^{10}-\frac{68\!\cdots\!77}{31\!\cdots\!71}a^{9}+\frac{18\!\cdots\!61}{31\!\cdots\!71}a^{8}-\frac{12\!\cdots\!02}{31\!\cdots\!71}a^{7}-\frac{32\!\cdots\!53}{31\!\cdots\!71}a^{6}+\frac{88\!\cdots\!04}{31\!\cdots\!71}a^{5}-\frac{21\!\cdots\!90}{31\!\cdots\!71}a^{4}-\frac{15\!\cdots\!89}{24\!\cdots\!67}a^{3}+\frac{36\!\cdots\!68}{31\!\cdots\!71}a^{2}+\frac{10\!\cdots\!99}{24\!\cdots\!67}a+\frac{40\!\cdots\!07}{31\!\cdots\!71}$, $\frac{11\!\cdots\!95}{31\!\cdots\!71}a^{19}-\frac{91\!\cdots\!39}{31\!\cdots\!71}a^{18}+\frac{31\!\cdots\!17}{31\!\cdots\!71}a^{17}-\frac{42\!\cdots\!35}{31\!\cdots\!71}a^{16}-\frac{55\!\cdots\!82}{24\!\cdots\!67}a^{15}+\frac{45\!\cdots\!13}{31\!\cdots\!71}a^{14}-\frac{94\!\cdots\!44}{31\!\cdots\!71}a^{13}+\frac{64\!\cdots\!06}{31\!\cdots\!71}a^{12}+\frac{15\!\cdots\!78}{31\!\cdots\!71}a^{11}-\frac{44\!\cdots\!62}{31\!\cdots\!71}a^{10}+\frac{41\!\cdots\!09}{31\!\cdots\!71}a^{9}+\frac{10\!\cdots\!10}{31\!\cdots\!71}a^{8}-\frac{52\!\cdots\!63}{31\!\cdots\!71}a^{7}+\frac{33\!\cdots\!37}{31\!\cdots\!71}a^{6}+\frac{80\!\cdots\!23}{31\!\cdots\!71}a^{5}-\frac{20\!\cdots\!89}{31\!\cdots\!71}a^{4}+\frac{49\!\cdots\!30}{24\!\cdots\!67}a^{3}+\frac{31\!\cdots\!77}{31\!\cdots\!71}a^{2}-\frac{55\!\cdots\!89}{24\!\cdots\!67}a-\frac{62\!\cdots\!36}{31\!\cdots\!71}$, $\frac{12\!\cdots\!26}{24\!\cdots\!67}a^{19}-\frac{63\!\cdots\!20}{24\!\cdots\!67}a^{18}+\frac{13\!\cdots\!71}{24\!\cdots\!67}a^{17}+\frac{47\!\cdots\!84}{24\!\cdots\!67}a^{16}-\frac{91\!\cdots\!64}{24\!\cdots\!67}a^{15}+\frac{22\!\cdots\!14}{24\!\cdots\!67}a^{14}-\frac{21\!\cdots\!85}{24\!\cdots\!67}a^{13}-\frac{35\!\cdots\!53}{24\!\cdots\!67}a^{12}+\frac{11\!\cdots\!91}{24\!\cdots\!67}a^{11}-\frac{10\!\cdots\!66}{24\!\cdots\!67}a^{10}-\frac{63\!\cdots\!07}{24\!\cdots\!67}a^{9}+\frac{17\!\cdots\!59}{24\!\cdots\!67}a^{8}-\frac{57\!\cdots\!31}{24\!\cdots\!67}a^{7}-\frac{69\!\cdots\!92}{24\!\cdots\!67}a^{6}+\frac{71\!\cdots\!32}{24\!\cdots\!67}a^{5}+\frac{97\!\cdots\!97}{24\!\cdots\!67}a^{4}-\frac{20\!\cdots\!36}{24\!\cdots\!67}a^{3}-\frac{28\!\cdots\!31}{24\!\cdots\!67}a^{2}+\frac{77\!\cdots\!33}{24\!\cdots\!67}a+\frac{29\!\cdots\!42}{24\!\cdots\!67}$, $\frac{21\!\cdots\!04}{24\!\cdots\!67}a^{19}-\frac{14\!\cdots\!87}{31\!\cdots\!71}a^{18}+\frac{33\!\cdots\!61}{31\!\cdots\!71}a^{17}+\frac{49\!\cdots\!33}{31\!\cdots\!71}a^{16}-\frac{20\!\cdots\!46}{31\!\cdots\!71}a^{15}+\frac{54\!\cdots\!73}{31\!\cdots\!71}a^{14}-\frac{57\!\cdots\!23}{31\!\cdots\!71}a^{13}-\frac{53\!\cdots\!45}{24\!\cdots\!67}a^{12}+\frac{28\!\cdots\!38}{31\!\cdots\!71}a^{11}-\frac{28\!\cdots\!33}{31\!\cdots\!71}a^{10}-\frac{95\!\cdots\!37}{31\!\cdots\!71}a^{9}+\frac{40\!\cdots\!95}{31\!\cdots\!71}a^{8}-\frac{18\!\cdots\!27}{31\!\cdots\!71}a^{7}-\frac{12\!\cdots\!56}{31\!\cdots\!71}a^{6}+\frac{17\!\cdots\!77}{31\!\cdots\!71}a^{5}-\frac{25\!\cdots\!64}{31\!\cdots\!71}a^{4}-\frac{42\!\cdots\!25}{31\!\cdots\!71}a^{3}-\frac{44\!\cdots\!82}{31\!\cdots\!71}a^{2}+\frac{22\!\cdots\!27}{31\!\cdots\!71}a+\frac{10\!\cdots\!81}{31\!\cdots\!71}$, $\frac{13\!\cdots\!88}{31\!\cdots\!71}a^{19}-\frac{16\!\cdots\!66}{31\!\cdots\!71}a^{18}-\frac{14\!\cdots\!78}{31\!\cdots\!71}a^{17}+\frac{75\!\cdots\!39}{31\!\cdots\!71}a^{16}-\frac{82\!\cdots\!35}{24\!\cdots\!67}a^{15}-\frac{14\!\cdots\!20}{31\!\cdots\!71}a^{14}+\frac{90\!\cdots\!33}{31\!\cdots\!71}a^{13}-\frac{17\!\cdots\!03}{31\!\cdots\!71}a^{12}+\frac{33\!\cdots\!99}{31\!\cdots\!71}a^{11}+\frac{43\!\cdots\!44}{31\!\cdots\!71}a^{10}-\frac{73\!\cdots\!10}{31\!\cdots\!71}a^{9}+\frac{17\!\cdots\!15}{31\!\cdots\!71}a^{8}+\frac{68\!\cdots\!34}{31\!\cdots\!71}a^{7}-\frac{59\!\cdots\!36}{31\!\cdots\!71}a^{6}-\frac{42\!\cdots\!34}{31\!\cdots\!71}a^{5}+\frac{35\!\cdots\!81}{31\!\cdots\!71}a^{4}-\frac{67\!\cdots\!25}{24\!\cdots\!67}a^{3}-\frac{79\!\cdots\!93}{31\!\cdots\!71}a^{2}+\frac{79\!\cdots\!63}{24\!\cdots\!67}a+\frac{69\!\cdots\!35}{31\!\cdots\!71}$, $\frac{22\!\cdots\!70}{31\!\cdots\!71}a^{19}-\frac{16\!\cdots\!05}{31\!\cdots\!71}a^{18}+\frac{52\!\cdots\!64}{31\!\cdots\!71}a^{17}-\frac{55\!\cdots\!95}{31\!\cdots\!71}a^{16}-\frac{13\!\cdots\!78}{24\!\cdots\!67}a^{15}+\frac{80\!\cdots\!67}{31\!\cdots\!71}a^{14}-\frac{14\!\cdots\!55}{31\!\cdots\!71}a^{13}+\frac{48\!\cdots\!47}{31\!\cdots\!71}a^{12}+\frac{34\!\cdots\!61}{31\!\cdots\!71}a^{11}-\frac{72\!\cdots\!18}{31\!\cdots\!71}a^{10}+\frac{43\!\cdots\!55}{31\!\cdots\!71}a^{9}+\frac{47\!\cdots\!60}{31\!\cdots\!71}a^{8}-\frac{87\!\cdots\!47}{31\!\cdots\!71}a^{7}+\frac{27\!\cdots\!64}{31\!\cdots\!71}a^{6}+\frac{33\!\cdots\!48}{31\!\cdots\!71}a^{5}-\frac{32\!\cdots\!86}{31\!\cdots\!71}a^{4}+\frac{97\!\cdots\!44}{24\!\cdots\!67}a^{3}+\frac{75\!\cdots\!09}{31\!\cdots\!71}a^{2}-\frac{30\!\cdots\!51}{24\!\cdots\!67}a-\frac{58\!\cdots\!22}{31\!\cdots\!71}$, $\frac{24\!\cdots\!68}{31\!\cdots\!71}a^{19}-\frac{12\!\cdots\!08}{31\!\cdots\!71}a^{18}+\frac{28\!\cdots\!17}{31\!\cdots\!71}a^{17}+\frac{75\!\cdots\!78}{31\!\cdots\!71}a^{16}-\frac{18\!\cdots\!93}{31\!\cdots\!71}a^{15}+\frac{47\!\cdots\!76}{31\!\cdots\!71}a^{14}-\frac{45\!\cdots\!76}{31\!\cdots\!71}a^{13}-\frac{69\!\cdots\!89}{31\!\cdots\!71}a^{12}+\frac{25\!\cdots\!69}{31\!\cdots\!71}a^{11}-\frac{22\!\cdots\!84}{31\!\cdots\!71}a^{10}-\frac{12\!\cdots\!60}{31\!\cdots\!71}a^{9}+\frac{38\!\cdots\!66}{31\!\cdots\!71}a^{8}-\frac{14\!\cdots\!18}{31\!\cdots\!71}a^{7}-\frac{15\!\cdots\!84}{31\!\cdots\!71}a^{6}+\frac{17\!\cdots\!60}{31\!\cdots\!71}a^{5}+\frac{10\!\cdots\!91}{31\!\cdots\!71}a^{4}-\frac{55\!\cdots\!44}{31\!\cdots\!71}a^{3}+\frac{79\!\cdots\!12}{31\!\cdots\!71}a^{2}+\frac{49\!\cdots\!14}{31\!\cdots\!71}a+\frac{27\!\cdots\!17}{31\!\cdots\!71}$, $\frac{10\!\cdots\!82}{31\!\cdots\!71}a^{19}-\frac{53\!\cdots\!30}{31\!\cdots\!71}a^{18}+\frac{11\!\cdots\!98}{31\!\cdots\!71}a^{17}+\frac{18\!\cdots\!85}{31\!\cdots\!71}a^{16}-\frac{74\!\cdots\!26}{31\!\cdots\!71}a^{15}+\frac{19\!\cdots\!12}{31\!\cdots\!71}a^{14}-\frac{20\!\cdots\!01}{31\!\cdots\!71}a^{13}-\frac{24\!\cdots\!55}{31\!\cdots\!71}a^{12}+\frac{10\!\cdots\!66}{31\!\cdots\!71}a^{11}-\frac{10\!\cdots\!69}{31\!\cdots\!71}a^{10}-\frac{32\!\cdots\!81}{31\!\cdots\!71}a^{9}+\frac{14\!\cdots\!68}{31\!\cdots\!71}a^{8}-\frac{69\!\cdots\!18}{31\!\cdots\!71}a^{7}-\frac{35\!\cdots\!53}{31\!\cdots\!71}a^{6}+\frac{55\!\cdots\!89}{31\!\cdots\!71}a^{5}-\frac{29\!\cdots\!40}{31\!\cdots\!71}a^{4}-\frac{97\!\cdots\!57}{31\!\cdots\!71}a^{3}-\frac{24\!\cdots\!03}{24\!\cdots\!67}a^{2}-\frac{16\!\cdots\!57}{31\!\cdots\!71}a+\frac{75\!\cdots\!38}{31\!\cdots\!71}$, $\frac{56\!\cdots\!46}{31\!\cdots\!71}a^{19}-\frac{29\!\cdots\!45}{31\!\cdots\!71}a^{18}+\frac{62\!\cdots\!07}{31\!\cdots\!71}a^{17}+\frac{21\!\cdots\!66}{31\!\cdots\!71}a^{16}-\frac{42\!\cdots\!61}{31\!\cdots\!71}a^{15}+\frac{10\!\cdots\!98}{31\!\cdots\!71}a^{14}-\frac{98\!\cdots\!69}{31\!\cdots\!71}a^{13}-\frac{16\!\cdots\!39}{31\!\cdots\!71}a^{12}+\frac{55\!\cdots\!59}{31\!\cdots\!71}a^{11}-\frac{48\!\cdots\!06}{31\!\cdots\!71}a^{10}-\frac{30\!\cdots\!09}{31\!\cdots\!71}a^{9}+\frac{82\!\cdots\!19}{31\!\cdots\!71}a^{8}-\frac{26\!\cdots\!22}{31\!\cdots\!71}a^{7}-\frac{34\!\cdots\!78}{31\!\cdots\!71}a^{6}+\frac{33\!\cdots\!94}{31\!\cdots\!71}a^{5}+\frac{53\!\cdots\!38}{31\!\cdots\!71}a^{4}-\frac{10\!\cdots\!01}{31\!\cdots\!71}a^{3}-\frac{16\!\cdots\!60}{31\!\cdots\!71}a^{2}+\frac{85\!\cdots\!43}{31\!\cdots\!71}a+\frac{18\!\cdots\!27}{31\!\cdots\!71}$, $\frac{25\!\cdots\!58}{31\!\cdots\!71}a^{19}-\frac{13\!\cdots\!65}{31\!\cdots\!71}a^{18}+\frac{30\!\cdots\!60}{31\!\cdots\!71}a^{17}+\frac{50\!\cdots\!26}{31\!\cdots\!71}a^{16}-\frac{19\!\cdots\!15}{31\!\cdots\!71}a^{15}+\frac{50\!\cdots\!13}{31\!\cdots\!71}a^{14}-\frac{51\!\cdots\!97}{31\!\cdots\!71}a^{13}-\frac{65\!\cdots\!56}{31\!\cdots\!71}a^{12}+\frac{25\!\cdots\!15}{31\!\cdots\!71}a^{11}-\frac{25\!\cdots\!04}{31\!\cdots\!71}a^{10}-\frac{97\!\cdots\!89}{31\!\cdots\!71}a^{9}+\frac{38\!\cdots\!69}{31\!\cdots\!71}a^{8}-\frac{17\!\cdots\!85}{31\!\cdots\!71}a^{7}-\frac{12\!\cdots\!38}{31\!\cdots\!71}a^{6}+\frac{17\!\cdots\!95}{31\!\cdots\!71}a^{5}-\frac{65\!\cdots\!31}{31\!\cdots\!71}a^{4}-\frac{45\!\cdots\!31}{31\!\cdots\!71}a^{3}+\frac{10\!\cdots\!04}{31\!\cdots\!71}a^{2}+\frac{24\!\cdots\!76}{31\!\cdots\!71}a+\frac{48\!\cdots\!60}{31\!\cdots\!71}$, $\frac{22\!\cdots\!18}{31\!\cdots\!71}a^{19}-\frac{11\!\cdots\!94}{31\!\cdots\!71}a^{18}+\frac{23\!\cdots\!83}{31\!\cdots\!71}a^{17}+\frac{82\!\cdots\!11}{24\!\cdots\!67}a^{16}-\frac{16\!\cdots\!71}{31\!\cdots\!71}a^{15}+\frac{39\!\cdots\!65}{31\!\cdots\!71}a^{14}-\frac{35\!\cdots\!37}{31\!\cdots\!71}a^{13}-\frac{65\!\cdots\!32}{31\!\cdots\!71}a^{12}+\frac{20\!\cdots\!12}{31\!\cdots\!71}a^{11}-\frac{12\!\cdots\!15}{24\!\cdots\!67}a^{10}-\frac{12\!\cdots\!56}{31\!\cdots\!71}a^{9}+\frac{28\!\cdots\!65}{31\!\cdots\!71}a^{8}-\frac{73\!\cdots\!05}{31\!\cdots\!71}a^{7}-\frac{11\!\cdots\!76}{31\!\cdots\!71}a^{6}+\frac{10\!\cdots\!46}{31\!\cdots\!71}a^{5}+\frac{26\!\cdots\!07}{31\!\cdots\!71}a^{4}-\frac{23\!\cdots\!24}{31\!\cdots\!71}a^{3}-\frac{94\!\cdots\!22}{31\!\cdots\!71}a^{2}-\frac{82\!\cdots\!61}{31\!\cdots\!71}a+\frac{42\!\cdots\!52}{31\!\cdots\!71}$
| 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: | \( 334840.411264 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 334840.411264 \cdot 1}{2\cdot\sqrt{240663579466141115896800721}}\cr\approx \mathstrut & 0.169989407659 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*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^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*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^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*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^20 - 5*x^19 + 10*x^18 + 6*x^17 - 74*x^16 + 171*x^15 - 136*x^14 - 322*x^13 + 921*x^12 - 653*x^11 - 699*x^10 + 1328*x^9 - 176*x^8 - 678*x^7 + 468*x^6 + 206*x^5 - 157*x^4 - 65*x^3 + 4*x^2 + 6*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
$C_2\wr S_5$ (as 20T288):
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 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 5.3.4511.1, 10.4.15513335536439.2, 10.4.7061144987.2, 10.6.44707018837.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.4.7061144987.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(347\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |