Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*x - 1)
gp: K = bnfinit(y^20 - 4*y^19 - 4*y^18 + 24*y^17 + 18*y^16 - 92*y^15 - 52*y^14 + 388*y^13 - 411*y^12 - 122*y^11 + 92*y^10 + 1352*y^9 - 1651*y^8 - 314*y^7 + 986*y^6 + 46*y^5 - 275*y^4 - 14*y^3 + 32*y^2 + 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*x - 1)
\( x^{20} - 4 x^{19} - 4 x^{18} + 24 x^{17} + 18 x^{16} - 92 x^{15} - 52 x^{14} + 388 x^{13} - 411 x^{12} + \cdots - 1 \)
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: | $[12, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(49338146756019243307761664\)
\(\medspace = 2^{30}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.26\) | 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: | $2^{31/16}11^{4/5}\approx 26.08313353110776$ | ||
| Ramified primes: |
\(2\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43}a^{18}-\frac{6}{43}a^{17}-\frac{10}{43}a^{16}-\frac{20}{43}a^{15}-\frac{20}{43}a^{14}+\frac{7}{43}a^{13}-\frac{7}{43}a^{12}+\frac{18}{43}a^{11}-\frac{20}{43}a^{10}-\frac{19}{43}a^{9}+\frac{17}{43}a^{8}-\frac{17}{43}a^{7}+\frac{12}{43}a^{6}+\frac{11}{43}a^{5}+\frac{17}{43}a^{4}-\frac{14}{43}a^{3}+\frac{6}{43}a^{2}+\frac{11}{43}a-\frac{12}{43}$, $\frac{1}{28\!\cdots\!81}a^{19}-\frac{11\!\cdots\!51}{28\!\cdots\!81}a^{18}+\frac{31\!\cdots\!59}{28\!\cdots\!81}a^{17}-\frac{81\!\cdots\!40}{28\!\cdots\!81}a^{16}+\frac{61\!\cdots\!12}{28\!\cdots\!81}a^{15}-\frac{11\!\cdots\!29}{28\!\cdots\!81}a^{14}+\frac{13\!\cdots\!99}{28\!\cdots\!81}a^{13}-\frac{94\!\cdots\!49}{28\!\cdots\!81}a^{12}+\frac{13\!\cdots\!23}{28\!\cdots\!81}a^{11}+\frac{97\!\cdots\!18}{28\!\cdots\!81}a^{10}-\frac{13\!\cdots\!64}{28\!\cdots\!81}a^{9}-\frac{32\!\cdots\!87}{28\!\cdots\!81}a^{8}+\frac{12\!\cdots\!29}{28\!\cdots\!81}a^{7}+\frac{26\!\cdots\!18}{66\!\cdots\!67}a^{6}-\frac{13\!\cdots\!63}{28\!\cdots\!81}a^{5}+\frac{32\!\cdots\!63}{28\!\cdots\!81}a^{4}-\frac{94\!\cdots\!59}{28\!\cdots\!81}a^{3}-\frac{12\!\cdots\!13}{28\!\cdots\!81}a^{2}-\frac{13\!\cdots\!76}{28\!\cdots\!81}a-\frac{62\!\cdots\!22}{28\!\cdots\!81}$
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: | $15$ | 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: |
$\frac{47\!\cdots\!56}{28\!\cdots\!81}a^{19}-\frac{20\!\cdots\!16}{28\!\cdots\!81}a^{18}-\frac{12\!\cdots\!84}{28\!\cdots\!81}a^{17}+\frac{11\!\cdots\!73}{28\!\cdots\!81}a^{16}+\frac{47\!\cdots\!20}{28\!\cdots\!81}a^{15}-\frac{44\!\cdots\!75}{28\!\cdots\!81}a^{14}-\frac{10\!\cdots\!23}{28\!\cdots\!81}a^{13}+\frac{18\!\cdots\!27}{28\!\cdots\!81}a^{12}-\frac{25\!\cdots\!27}{28\!\cdots\!81}a^{11}+\frac{25\!\cdots\!83}{28\!\cdots\!81}a^{10}+\frac{31\!\cdots\!07}{28\!\cdots\!81}a^{9}+\frac{62\!\cdots\!85}{28\!\cdots\!81}a^{8}-\frac{96\!\cdots\!90}{28\!\cdots\!81}a^{7}+\frac{17\!\cdots\!40}{28\!\cdots\!81}a^{6}+\frac{39\!\cdots\!12}{28\!\cdots\!81}a^{5}-\frac{11\!\cdots\!20}{28\!\cdots\!81}a^{4}-\frac{89\!\cdots\!74}{28\!\cdots\!81}a^{3}+\frac{23\!\cdots\!18}{28\!\cdots\!81}a^{2}+\frac{69\!\cdots\!29}{28\!\cdots\!81}a-\frac{14\!\cdots\!65}{28\!\cdots\!81}$, $\frac{22\!\cdots\!39}{28\!\cdots\!81}a^{19}-\frac{98\!\cdots\!53}{28\!\cdots\!81}a^{18}-\frac{43\!\cdots\!31}{28\!\cdots\!81}a^{17}+\frac{54\!\cdots\!76}{28\!\cdots\!81}a^{16}+\frac{12\!\cdots\!17}{28\!\cdots\!81}a^{15}-\frac{20\!\cdots\!03}{28\!\cdots\!81}a^{14}-\frac{37\!\cdots\!82}{28\!\cdots\!81}a^{13}+\frac{87\!\cdots\!49}{28\!\cdots\!81}a^{12}-\frac{13\!\cdots\!36}{28\!\cdots\!81}a^{11}+\frac{35\!\cdots\!26}{28\!\cdots\!81}a^{10}+\frac{11\!\cdots\!52}{28\!\cdots\!81}a^{9}+\frac{28\!\cdots\!53}{28\!\cdots\!81}a^{8}-\frac{11\!\cdots\!99}{66\!\cdots\!67}a^{7}+\frac{17\!\cdots\!29}{28\!\cdots\!81}a^{6}+\frac{18\!\cdots\!64}{28\!\cdots\!81}a^{5}-\frac{92\!\cdots\!00}{28\!\cdots\!81}a^{4}-\frac{43\!\cdots\!84}{28\!\cdots\!81}a^{3}+\frac{19\!\cdots\!30}{28\!\cdots\!81}a^{2}+\frac{41\!\cdots\!14}{28\!\cdots\!81}a-\frac{11\!\cdots\!22}{28\!\cdots\!81}$, $\frac{35\!\cdots\!60}{28\!\cdots\!81}a^{19}-\frac{15\!\cdots\!09}{28\!\cdots\!81}a^{18}-\frac{90\!\cdots\!78}{28\!\cdots\!81}a^{17}+\frac{88\!\cdots\!32}{28\!\cdots\!81}a^{16}+\frac{34\!\cdots\!29}{28\!\cdots\!81}a^{15}-\frac{33\!\cdots\!73}{28\!\cdots\!81}a^{14}-\frac{71\!\cdots\!11}{28\!\cdots\!81}a^{13}+\frac{14\!\cdots\!49}{28\!\cdots\!81}a^{12}-\frac{19\!\cdots\!06}{28\!\cdots\!81}a^{11}+\frac{20\!\cdots\!79}{28\!\cdots\!81}a^{10}+\frac{26\!\cdots\!52}{28\!\cdots\!81}a^{9}+\frac{47\!\cdots\!40}{28\!\cdots\!81}a^{8}-\frac{74\!\cdots\!21}{28\!\cdots\!81}a^{7}+\frac{13\!\cdots\!34}{28\!\cdots\!81}a^{6}+\frac{30\!\cdots\!37}{28\!\cdots\!81}a^{5}-\frac{84\!\cdots\!97}{28\!\cdots\!81}a^{4}-\frac{70\!\cdots\!00}{28\!\cdots\!81}a^{3}+\frac{18\!\cdots\!57}{28\!\cdots\!81}a^{2}+\frac{53\!\cdots\!61}{28\!\cdots\!81}a-\frac{10\!\cdots\!12}{28\!\cdots\!81}$, $\frac{37\!\cdots\!67}{28\!\cdots\!81}a^{19}-\frac{16\!\cdots\!33}{28\!\cdots\!81}a^{18}-\frac{93\!\cdots\!06}{28\!\cdots\!81}a^{17}+\frac{92\!\cdots\!27}{28\!\cdots\!81}a^{16}+\frac{35\!\cdots\!79}{28\!\cdots\!81}a^{15}-\frac{35\!\cdots\!71}{28\!\cdots\!81}a^{14}-\frac{71\!\cdots\!58}{28\!\cdots\!81}a^{13}+\frac{14\!\cdots\!72}{28\!\cdots\!81}a^{12}-\frac{20\!\cdots\!25}{28\!\cdots\!81}a^{11}+\frac{23\!\cdots\!79}{28\!\cdots\!81}a^{10}+\frac{27\!\cdots\!84}{28\!\cdots\!81}a^{9}+\frac{49\!\cdots\!47}{28\!\cdots\!81}a^{8}-\frac{18\!\cdots\!01}{66\!\cdots\!67}a^{7}+\frac{14\!\cdots\!27}{28\!\cdots\!81}a^{6}+\frac{32\!\cdots\!47}{28\!\cdots\!81}a^{5}-\frac{92\!\cdots\!03}{28\!\cdots\!81}a^{4}-\frac{74\!\cdots\!33}{28\!\cdots\!81}a^{3}+\frac{19\!\cdots\!99}{28\!\cdots\!81}a^{2}+\frac{57\!\cdots\!86}{28\!\cdots\!81}a-\frac{11\!\cdots\!40}{28\!\cdots\!81}$, $\frac{48\!\cdots\!59}{28\!\cdots\!81}a^{19}-\frac{21\!\cdots\!94}{28\!\cdots\!81}a^{18}-\frac{12\!\cdots\!52}{28\!\cdots\!81}a^{17}+\frac{12\!\cdots\!19}{28\!\cdots\!81}a^{16}+\frac{49\!\cdots\!78}{28\!\cdots\!81}a^{15}-\frac{48\!\cdots\!76}{28\!\cdots\!81}a^{14}-\frac{11\!\cdots\!38}{28\!\cdots\!81}a^{13}+\frac{19\!\cdots\!32}{28\!\cdots\!81}a^{12}-\frac{26\!\cdots\!95}{28\!\cdots\!81}a^{11}+\frac{13\!\cdots\!93}{28\!\cdots\!81}a^{10}+\frac{49\!\cdots\!97}{28\!\cdots\!81}a^{9}+\frac{66\!\cdots\!30}{28\!\cdots\!81}a^{8}-\frac{23\!\cdots\!38}{66\!\cdots\!67}a^{7}+\frac{12\!\cdots\!16}{28\!\cdots\!81}a^{6}+\frac{45\!\cdots\!66}{28\!\cdots\!81}a^{5}-\frac{88\!\cdots\!68}{28\!\cdots\!81}a^{4}-\frac{10\!\cdots\!13}{28\!\cdots\!81}a^{3}+\frac{17\!\cdots\!40}{28\!\cdots\!81}a^{2}+\frac{93\!\cdots\!78}{28\!\cdots\!81}a-\frac{87\!\cdots\!62}{28\!\cdots\!81}$, $\frac{18\!\cdots\!16}{28\!\cdots\!81}a^{19}-\frac{79\!\cdots\!85}{28\!\cdots\!81}a^{18}-\frac{50\!\cdots\!21}{28\!\cdots\!81}a^{17}+\frac{45\!\cdots\!33}{28\!\cdots\!81}a^{16}+\frac{19\!\cdots\!74}{28\!\cdots\!81}a^{15}-\frac{17\!\cdots\!40}{28\!\cdots\!81}a^{14}-\frac{42\!\cdots\!10}{28\!\cdots\!81}a^{13}+\frac{72\!\cdots\!65}{28\!\cdots\!81}a^{12}-\frac{98\!\cdots\!02}{28\!\cdots\!81}a^{11}+\frac{91\!\cdots\!36}{28\!\cdots\!81}a^{10}+\frac{11\!\cdots\!72}{28\!\cdots\!81}a^{9}+\frac{24\!\cdots\!76}{28\!\cdots\!81}a^{8}-\frac{37\!\cdots\!20}{28\!\cdots\!81}a^{7}+\frac{64\!\cdots\!48}{28\!\cdots\!81}a^{6}+\frac{15\!\cdots\!49}{28\!\cdots\!81}a^{5}-\frac{41\!\cdots\!84}{28\!\cdots\!81}a^{4}-\frac{35\!\cdots\!19}{28\!\cdots\!81}a^{3}+\frac{88\!\cdots\!83}{28\!\cdots\!81}a^{2}+\frac{27\!\cdots\!82}{28\!\cdots\!81}a-\frac{51\!\cdots\!75}{28\!\cdots\!81}$, $\frac{33\!\cdots\!05}{28\!\cdots\!81}a^{19}-\frac{14\!\cdots\!64}{28\!\cdots\!81}a^{18}-\frac{81\!\cdots\!94}{28\!\cdots\!81}a^{17}+\frac{83\!\cdots\!93}{28\!\cdots\!81}a^{16}+\frac{30\!\cdots\!16}{28\!\cdots\!81}a^{15}-\frac{31\!\cdots\!41}{28\!\cdots\!81}a^{14}-\frac{60\!\cdots\!37}{28\!\cdots\!81}a^{13}+\frac{13\!\cdots\!47}{28\!\cdots\!81}a^{12}-\frac{18\!\cdots\!14}{28\!\cdots\!81}a^{11}+\frac{22\!\cdots\!22}{28\!\cdots\!81}a^{10}+\frac{24\!\cdots\!62}{28\!\cdots\!81}a^{9}+\frac{44\!\cdots\!85}{28\!\cdots\!81}a^{8}-\frac{70\!\cdots\!84}{28\!\cdots\!81}a^{7}+\frac{14\!\cdots\!01}{28\!\cdots\!81}a^{6}+\frac{28\!\cdots\!75}{28\!\cdots\!81}a^{5}-\frac{85\!\cdots\!76}{28\!\cdots\!81}a^{4}-\frac{65\!\cdots\!96}{28\!\cdots\!81}a^{3}+\frac{18\!\cdots\!93}{28\!\cdots\!81}a^{2}+\frac{50\!\cdots\!89}{28\!\cdots\!81}a-\frac{10\!\cdots\!17}{28\!\cdots\!81}$, $\frac{48\!\cdots\!08}{28\!\cdots\!81}a^{19}-\frac{21\!\cdots\!74}{28\!\cdots\!81}a^{18}-\frac{11\!\cdots\!00}{28\!\cdots\!81}a^{17}+\frac{12\!\cdots\!41}{28\!\cdots\!81}a^{16}+\frac{42\!\cdots\!96}{28\!\cdots\!81}a^{15}-\frac{46\!\cdots\!05}{28\!\cdots\!81}a^{14}-\frac{83\!\cdots\!82}{28\!\cdots\!81}a^{13}+\frac{19\!\cdots\!74}{28\!\cdots\!81}a^{12}-\frac{27\!\cdots\!09}{28\!\cdots\!81}a^{11}+\frac{33\!\cdots\!48}{28\!\cdots\!81}a^{10}+\frac{39\!\cdots\!84}{28\!\cdots\!81}a^{9}+\frac{64\!\cdots\!96}{28\!\cdots\!81}a^{8}-\frac{10\!\cdots\!61}{28\!\cdots\!81}a^{7}+\frac{20\!\cdots\!63}{28\!\cdots\!81}a^{6}+\frac{43\!\cdots\!70}{28\!\cdots\!81}a^{5}-\frac{29\!\cdots\!52}{66\!\cdots\!67}a^{4}-\frac{99\!\cdots\!01}{28\!\cdots\!81}a^{3}+\frac{26\!\cdots\!60}{28\!\cdots\!81}a^{2}+\frac{77\!\cdots\!63}{28\!\cdots\!81}a-\frac{15\!\cdots\!62}{28\!\cdots\!81}$, $\frac{81\!\cdots\!53}{28\!\cdots\!81}a^{19}-\frac{54\!\cdots\!36}{28\!\cdots\!81}a^{18}+\frac{36\!\cdots\!64}{28\!\cdots\!81}a^{17}+\frac{32\!\cdots\!10}{28\!\cdots\!81}a^{16}-\frac{22\!\cdots\!48}{28\!\cdots\!81}a^{15}-\frac{13\!\cdots\!33}{28\!\cdots\!81}a^{14}+\frac{85\!\cdots\!04}{28\!\cdots\!81}a^{13}+\frac{48\!\cdots\!10}{28\!\cdots\!81}a^{12}-\frac{96\!\cdots\!92}{28\!\cdots\!81}a^{11}+\frac{42\!\cdots\!78}{28\!\cdots\!81}a^{10}+\frac{36\!\cdots\!72}{28\!\cdots\!81}a^{9}+\frac{13\!\cdots\!67}{28\!\cdots\!81}a^{8}-\frac{37\!\cdots\!28}{28\!\cdots\!81}a^{7}+\frac{16\!\cdots\!60}{28\!\cdots\!81}a^{6}+\frac{14\!\cdots\!51}{28\!\cdots\!81}a^{5}-\frac{77\!\cdots\!50}{28\!\cdots\!81}a^{4}-\frac{32\!\cdots\!10}{28\!\cdots\!81}a^{3}+\frac{14\!\cdots\!73}{28\!\cdots\!81}a^{2}+\frac{27\!\cdots\!53}{28\!\cdots\!81}a-\frac{58\!\cdots\!45}{28\!\cdots\!81}$, $\frac{28\!\cdots\!98}{28\!\cdots\!81}a^{19}-\frac{12\!\cdots\!14}{28\!\cdots\!81}a^{18}-\frac{69\!\cdots\!34}{28\!\cdots\!81}a^{17}+\frac{70\!\cdots\!38}{28\!\cdots\!81}a^{16}+\frac{26\!\cdots\!51}{28\!\cdots\!81}a^{15}-\frac{27\!\cdots\!63}{28\!\cdots\!81}a^{14}-\frac{53\!\cdots\!55}{28\!\cdots\!81}a^{13}+\frac{11\!\cdots\!49}{28\!\cdots\!81}a^{12}-\frac{15\!\cdots\!16}{28\!\cdots\!81}a^{11}+\frac{16\!\cdots\!54}{28\!\cdots\!81}a^{10}+\frac{23\!\cdots\!80}{28\!\cdots\!81}a^{9}+\frac{37\!\cdots\!10}{28\!\cdots\!81}a^{8}-\frac{59\!\cdots\!43}{28\!\cdots\!81}a^{7}+\frac{10\!\cdots\!27}{28\!\cdots\!81}a^{6}+\frac{24\!\cdots\!95}{28\!\cdots\!81}a^{5}-\frac{67\!\cdots\!83}{28\!\cdots\!81}a^{4}-\frac{56\!\cdots\!52}{28\!\cdots\!81}a^{3}+\frac{33\!\cdots\!84}{66\!\cdots\!67}a^{2}+\frac{43\!\cdots\!16}{28\!\cdots\!81}a-\frac{85\!\cdots\!97}{28\!\cdots\!81}$, $\frac{42\!\cdots\!75}{28\!\cdots\!81}a^{19}-\frac{18\!\cdots\!95}{28\!\cdots\!81}a^{18}-\frac{10\!\cdots\!42}{28\!\cdots\!81}a^{17}+\frac{10\!\cdots\!08}{28\!\cdots\!81}a^{16}+\frac{94\!\cdots\!40}{66\!\cdots\!67}a^{15}-\frac{40\!\cdots\!77}{28\!\cdots\!81}a^{14}-\frac{84\!\cdots\!78}{28\!\cdots\!81}a^{13}+\frac{16\!\cdots\!41}{28\!\cdots\!81}a^{12}-\frac{22\!\cdots\!94}{28\!\cdots\!81}a^{11}+\frac{25\!\cdots\!72}{28\!\cdots\!81}a^{10}+\frac{30\!\cdots\!49}{28\!\cdots\!81}a^{9}+\frac{55\!\cdots\!27}{28\!\cdots\!81}a^{8}-\frac{88\!\cdots\!50}{28\!\cdots\!81}a^{7}+\frac{16\!\cdots\!89}{28\!\cdots\!81}a^{6}+\frac{36\!\cdots\!74}{28\!\cdots\!81}a^{5}-\frac{10\!\cdots\!95}{28\!\cdots\!81}a^{4}-\frac{82\!\cdots\!72}{28\!\cdots\!81}a^{3}+\frac{22\!\cdots\!13}{28\!\cdots\!81}a^{2}+\frac{63\!\cdots\!27}{28\!\cdots\!81}a-\frac{13\!\cdots\!57}{28\!\cdots\!81}$, $\frac{87\!\cdots\!00}{28\!\cdots\!81}a^{19}-\frac{39\!\cdots\!02}{28\!\cdots\!81}a^{18}-\frac{38\!\cdots\!36}{66\!\cdots\!67}a^{17}+\frac{22\!\cdots\!12}{28\!\cdots\!81}a^{16}+\frac{53\!\cdots\!73}{28\!\cdots\!81}a^{15}-\frac{87\!\cdots\!42}{28\!\cdots\!81}a^{14}-\frac{70\!\cdots\!62}{28\!\cdots\!81}a^{13}+\frac{35\!\cdots\!74}{28\!\cdots\!81}a^{12}-\frac{52\!\cdots\!86}{28\!\cdots\!81}a^{11}+\frac{22\!\cdots\!58}{66\!\cdots\!67}a^{10}+\frac{82\!\cdots\!20}{28\!\cdots\!81}a^{9}+\frac{11\!\cdots\!96}{28\!\cdots\!81}a^{8}-\frac{20\!\cdots\!08}{28\!\cdots\!81}a^{7}+\frac{49\!\cdots\!87}{28\!\cdots\!81}a^{6}+\frac{78\!\cdots\!18}{28\!\cdots\!81}a^{5}-\frac{26\!\cdots\!63}{28\!\cdots\!81}a^{4}-\frac{18\!\cdots\!33}{28\!\cdots\!81}a^{3}+\frac{54\!\cdots\!20}{28\!\cdots\!81}a^{2}+\frac{13\!\cdots\!77}{28\!\cdots\!81}a-\frac{31\!\cdots\!89}{28\!\cdots\!81}$, $\frac{13\!\cdots\!97}{28\!\cdots\!81}a^{19}-\frac{60\!\cdots\!01}{28\!\cdots\!81}a^{18}-\frac{33\!\cdots\!98}{28\!\cdots\!81}a^{17}+\frac{35\!\cdots\!63}{28\!\cdots\!81}a^{16}+\frac{12\!\cdots\!65}{28\!\cdots\!81}a^{15}-\frac{13\!\cdots\!51}{28\!\cdots\!81}a^{14}-\frac{26\!\cdots\!30}{28\!\cdots\!81}a^{13}+\frac{55\!\cdots\!54}{28\!\cdots\!81}a^{12}-\frac{76\!\cdots\!89}{28\!\cdots\!81}a^{11}+\frac{71\!\cdots\!37}{28\!\cdots\!81}a^{10}+\frac{13\!\cdots\!66}{28\!\cdots\!81}a^{9}+\frac{18\!\cdots\!62}{28\!\cdots\!81}a^{8}-\frac{29\!\cdots\!99}{28\!\cdots\!81}a^{7}+\frac{47\!\cdots\!40}{28\!\cdots\!81}a^{6}+\frac{12\!\cdots\!08}{28\!\cdots\!81}a^{5}-\frac{30\!\cdots\!30}{28\!\cdots\!81}a^{4}-\frac{29\!\cdots\!70}{28\!\cdots\!81}a^{3}+\frac{60\!\cdots\!68}{28\!\cdots\!81}a^{2}+\frac{22\!\cdots\!48}{28\!\cdots\!81}a-\frac{34\!\cdots\!88}{28\!\cdots\!81}$, $\frac{26\!\cdots\!48}{28\!\cdots\!81}a^{19}-\frac{11\!\cdots\!31}{28\!\cdots\!81}a^{18}-\frac{61\!\cdots\!96}{28\!\cdots\!81}a^{17}+\frac{66\!\cdots\!44}{28\!\cdots\!81}a^{16}+\frac{22\!\cdots\!76}{28\!\cdots\!81}a^{15}-\frac{25\!\cdots\!01}{28\!\cdots\!81}a^{14}-\frac{42\!\cdots\!32}{28\!\cdots\!81}a^{13}+\frac{10\!\cdots\!75}{28\!\cdots\!81}a^{12}-\frac{14\!\cdots\!70}{28\!\cdots\!81}a^{11}+\frac{20\!\cdots\!32}{28\!\cdots\!81}a^{10}+\frac{21\!\cdots\!57}{28\!\cdots\!81}a^{9}+\frac{35\!\cdots\!52}{28\!\cdots\!81}a^{8}-\frac{57\!\cdots\!77}{28\!\cdots\!81}a^{7}+\frac{11\!\cdots\!81}{28\!\cdots\!81}a^{6}+\frac{23\!\cdots\!90}{28\!\cdots\!81}a^{5}-\frac{71\!\cdots\!18}{28\!\cdots\!81}a^{4}-\frac{52\!\cdots\!68}{28\!\cdots\!81}a^{3}+\frac{15\!\cdots\!69}{28\!\cdots\!81}a^{2}+\frac{40\!\cdots\!11}{28\!\cdots\!81}a-\frac{21\!\cdots\!70}{66\!\cdots\!67}$, $\frac{50\!\cdots\!47}{28\!\cdots\!81}a^{19}-\frac{22\!\cdots\!73}{28\!\cdots\!81}a^{18}-\frac{12\!\cdots\!81}{28\!\cdots\!81}a^{17}+\frac{12\!\cdots\!77}{28\!\cdots\!81}a^{16}+\frac{46\!\cdots\!92}{28\!\cdots\!81}a^{15}-\frac{48\!\cdots\!86}{28\!\cdots\!81}a^{14}-\frac{93\!\cdots\!07}{28\!\cdots\!81}a^{13}+\frac{20\!\cdots\!78}{28\!\cdots\!81}a^{12}-\frac{28\!\cdots\!06}{28\!\cdots\!81}a^{11}+\frac{33\!\cdots\!09}{28\!\cdots\!81}a^{10}+\frac{39\!\cdots\!99}{28\!\cdots\!81}a^{9}+\frac{67\!\cdots\!10}{28\!\cdots\!81}a^{8}-\frac{10\!\cdots\!10}{28\!\cdots\!81}a^{7}+\frac{20\!\cdots\!83}{28\!\cdots\!81}a^{6}+\frac{44\!\cdots\!84}{28\!\cdots\!81}a^{5}-\frac{12\!\cdots\!88}{28\!\cdots\!81}a^{4}-\frac{10\!\cdots\!51}{28\!\cdots\!81}a^{3}+\frac{26\!\cdots\!69}{28\!\cdots\!81}a^{2}+\frac{78\!\cdots\!37}{28\!\cdots\!81}a-\frac{15\!\cdots\!03}{28\!\cdots\!81}$
| 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: | \( 322822.36902 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{4}\cdot 322822.36902 \cdot 1}{2\cdot\sqrt{49338146756019243307761664}}\cr\approx \mathstrut & 0.14669713419 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*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 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*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 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*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 - 4*x^19 - 4*x^18 + 24*x^17 + 18*x^16 - 92*x^15 - 52*x^14 + 388*x^13 - 411*x^12 - 122*x^11 + 92*x^10 + 1352*x^9 - 1651*x^8 - 314*x^7 + 986*x^6 + 46*x^5 - 275*x^4 - 14*x^3 + 32*x^2 + 2*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
$C_2\wr C_5$ (as 20T41):
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 solvable group of order 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.8.219503494144.1 x2, 10.6.219503494144.1 |
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 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.8.219503494144.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $30$ | |||
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |