Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033)
gp: K = bnfinit(y^20 - y^19 - 49*y^18 + 32*y^17 + 926*y^16 - 354*y^15 - 8765*y^14 + 1660*y^13 + 45828*y^12 - 2629*y^11 - 137522*y^10 - 4409*y^9 + 237299*y^8 + 22455*y^7 - 226443*y^6 - 31515*y^5 + 107725*y^4 + 17397*y^3 - 20309*y^2 - 2449*y + 1033, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033)
\( x^{20} - x^{19} - 49 x^{18} + 32 x^{17} + 926 x^{16} - 354 x^{15} - 8765 x^{14} + 1660 x^{13} + \cdots + 1033 \)
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: | $[20, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(131527565972137936816816034072938673\)
\(\medspace = 11^{16}\cdot 17^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.01\) | 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: | $11^{4/5}17^{3/4}\approx 57.00997350388851$ | ||
| Ramified primes: |
\(11\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(187=11\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{187}(64,·)$, $\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(4,·)$, $\chi_{187}(69,·)$, $\chi_{187}(135,·)$, $\chi_{187}(137,·)$, $\chi_{187}(16,·)$, $\chi_{187}(81,·)$, $\chi_{187}(86,·)$, $\chi_{187}(152,·)$, $\chi_{187}(89,·)$, $\chi_{187}(157,·)$, $\chi_{187}(166,·)$, $\chi_{187}(38,·)$, $\chi_{187}(103,·)$, $\chi_{187}(169,·)$, $\chi_{187}(174,·)$, $\chi_{187}(47,·)$, $\chi_{187}(115,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{22\!\cdots\!66}a^{19}-\frac{39\!\cdots\!61}{22\!\cdots\!66}a^{18}+\frac{27\!\cdots\!43}{11\!\cdots\!33}a^{17}-\frac{58\!\cdots\!29}{22\!\cdots\!66}a^{16}-\frac{37\!\cdots\!55}{22\!\cdots\!66}a^{15}+\frac{34\!\cdots\!68}{11\!\cdots\!33}a^{14}-\frac{58\!\cdots\!69}{22\!\cdots\!66}a^{13}+\frac{88\!\cdots\!33}{22\!\cdots\!66}a^{12}-\frac{11\!\cdots\!59}{11\!\cdots\!33}a^{11}+\frac{69\!\cdots\!71}{11\!\cdots\!33}a^{10}-\frac{57\!\cdots\!67}{22\!\cdots\!66}a^{9}+\frac{62\!\cdots\!01}{22\!\cdots\!66}a^{8}+\frac{21\!\cdots\!67}{22\!\cdots\!66}a^{7}+\frac{34\!\cdots\!33}{22\!\cdots\!66}a^{6}+\frac{10\!\cdots\!31}{11\!\cdots\!33}a^{5}-\frac{89\!\cdots\!79}{22\!\cdots\!66}a^{4}-\frac{90\!\cdots\!03}{22\!\cdots\!66}a^{3}+\frac{32\!\cdots\!07}{11\!\cdots\!33}a^{2}+\frac{45\!\cdots\!89}{22\!\cdots\!66}a-\frac{20\!\cdots\!13}{11\!\cdots\!33}$
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: | $19$ | 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: |
$\frac{28\!\cdots\!86}{11\!\cdots\!33}a^{19}+\frac{48\!\cdots\!33}{11\!\cdots\!33}a^{18}-\frac{14\!\cdots\!11}{11\!\cdots\!33}a^{17}-\frac{69\!\cdots\!81}{11\!\cdots\!33}a^{16}+\frac{26\!\cdots\!81}{11\!\cdots\!33}a^{15}+\frac{18\!\cdots\!53}{11\!\cdots\!33}a^{14}-\frac{23\!\cdots\!06}{11\!\cdots\!33}a^{13}-\frac{20\!\cdots\!16}{11\!\cdots\!33}a^{12}+\frac{11\!\cdots\!67}{11\!\cdots\!33}a^{11}+\frac{10\!\cdots\!08}{11\!\cdots\!33}a^{10}-\frac{29\!\cdots\!73}{11\!\cdots\!33}a^{9}-\frac{26\!\cdots\!06}{11\!\cdots\!33}a^{8}+\frac{39\!\cdots\!43}{11\!\cdots\!33}a^{7}+\frac{34\!\cdots\!34}{11\!\cdots\!33}a^{6}-\frac{24\!\cdots\!63}{11\!\cdots\!33}a^{5}-\frac{19\!\cdots\!03}{11\!\cdots\!33}a^{4}+\frac{49\!\cdots\!14}{11\!\cdots\!33}a^{3}+\frac{41\!\cdots\!48}{11\!\cdots\!33}a^{2}-\frac{73\!\cdots\!46}{11\!\cdots\!33}a-\frac{24\!\cdots\!76}{11\!\cdots\!33}$, $\frac{11\!\cdots\!10}{11\!\cdots\!33}a^{19}-\frac{59\!\cdots\!28}{11\!\cdots\!33}a^{18}-\frac{54\!\cdots\!98}{11\!\cdots\!33}a^{17}+\frac{78\!\cdots\!29}{11\!\cdots\!33}a^{16}+\frac{94\!\cdots\!72}{11\!\cdots\!33}a^{15}+\frac{13\!\cdots\!90}{11\!\cdots\!33}a^{14}-\frac{76\!\cdots\!02}{11\!\cdots\!33}a^{13}-\frac{26\!\cdots\!34}{11\!\cdots\!33}a^{12}+\frac{30\!\cdots\!74}{11\!\cdots\!33}a^{11}+\frac{15\!\cdots\!94}{11\!\cdots\!33}a^{10}-\frac{55\!\cdots\!68}{11\!\cdots\!33}a^{9}-\frac{35\!\cdots\!67}{11\!\cdots\!33}a^{8}+\frac{23\!\cdots\!36}{11\!\cdots\!33}a^{7}+\frac{34\!\cdots\!46}{11\!\cdots\!33}a^{6}+\frac{42\!\cdots\!08}{11\!\cdots\!33}a^{5}-\frac{80\!\cdots\!89}{11\!\cdots\!33}a^{4}-\frac{41\!\cdots\!70}{11\!\cdots\!33}a^{3}-\frac{41\!\cdots\!05}{11\!\cdots\!33}a^{2}+\frac{73\!\cdots\!36}{11\!\cdots\!33}a-\frac{81\!\cdots\!73}{11\!\cdots\!33}$, $\frac{25\!\cdots\!20}{11\!\cdots\!33}a^{19}+\frac{14\!\cdots\!54}{11\!\cdots\!33}a^{18}-\frac{12\!\cdots\!08}{11\!\cdots\!33}a^{17}-\frac{10\!\cdots\!69}{11\!\cdots\!33}a^{16}+\frac{22\!\cdots\!57}{11\!\cdots\!33}a^{15}+\frac{25\!\cdots\!17}{11\!\cdots\!33}a^{14}-\frac{19\!\cdots\!39}{11\!\cdots\!33}a^{13}-\frac{26\!\cdots\!30}{11\!\cdots\!33}a^{12}+\frac{91\!\cdots\!96}{11\!\cdots\!33}a^{11}+\frac{13\!\cdots\!83}{11\!\cdots\!33}a^{10}-\frac{22\!\cdots\!79}{11\!\cdots\!33}a^{9}-\frac{34\!\cdots\!34}{11\!\cdots\!33}a^{8}+\frac{27\!\cdots\!81}{11\!\cdots\!33}a^{7}+\frac{46\!\cdots\!71}{11\!\cdots\!33}a^{6}-\frac{15\!\cdots\!25}{11\!\cdots\!33}a^{5}-\frac{28\!\cdots\!59}{11\!\cdots\!33}a^{4}+\frac{29\!\cdots\!89}{11\!\cdots\!33}a^{3}+\frac{66\!\cdots\!50}{11\!\cdots\!33}a^{2}+\frac{68\!\cdots\!90}{11\!\cdots\!33}a-\frac{35\!\cdots\!15}{11\!\cdots\!33}$, $\frac{11\!\cdots\!10}{11\!\cdots\!33}a^{19}-\frac{59\!\cdots\!28}{11\!\cdots\!33}a^{18}-\frac{54\!\cdots\!98}{11\!\cdots\!33}a^{17}+\frac{78\!\cdots\!29}{11\!\cdots\!33}a^{16}+\frac{94\!\cdots\!72}{11\!\cdots\!33}a^{15}+\frac{13\!\cdots\!90}{11\!\cdots\!33}a^{14}-\frac{76\!\cdots\!02}{11\!\cdots\!33}a^{13}-\frac{26\!\cdots\!34}{11\!\cdots\!33}a^{12}+\frac{30\!\cdots\!74}{11\!\cdots\!33}a^{11}+\frac{15\!\cdots\!94}{11\!\cdots\!33}a^{10}-\frac{55\!\cdots\!68}{11\!\cdots\!33}a^{9}-\frac{35\!\cdots\!67}{11\!\cdots\!33}a^{8}+\frac{23\!\cdots\!36}{11\!\cdots\!33}a^{7}+\frac{34\!\cdots\!46}{11\!\cdots\!33}a^{6}+\frac{42\!\cdots\!08}{11\!\cdots\!33}a^{5}-\frac{80\!\cdots\!89}{11\!\cdots\!33}a^{4}-\frac{41\!\cdots\!70}{11\!\cdots\!33}a^{3}-\frac{41\!\cdots\!05}{11\!\cdots\!33}a^{2}+\frac{73\!\cdots\!36}{11\!\cdots\!33}a-\frac{19\!\cdots\!06}{11\!\cdots\!33}$, $\frac{38\!\cdots\!63}{11\!\cdots\!33}a^{19}-\frac{15\!\cdots\!29}{22\!\cdots\!66}a^{18}-\frac{37\!\cdots\!63}{22\!\cdots\!66}a^{17}-\frac{49\!\cdots\!49}{22\!\cdots\!66}a^{16}+\frac{34\!\cdots\!28}{11\!\cdots\!33}a^{15}+\frac{12\!\cdots\!83}{11\!\cdots\!33}a^{14}-\frac{61\!\cdots\!83}{22\!\cdots\!66}a^{13}-\frac{15\!\cdots\!84}{11\!\cdots\!33}a^{12}+\frac{29\!\cdots\!75}{22\!\cdots\!66}a^{11}+\frac{16\!\cdots\!25}{22\!\cdots\!66}a^{10}-\frac{38\!\cdots\!75}{11\!\cdots\!33}a^{9}-\frac{20\!\cdots\!33}{11\!\cdots\!33}a^{8}+\frac{54\!\cdots\!41}{11\!\cdots\!33}a^{7}+\frac{45\!\cdots\!85}{22\!\cdots\!66}a^{6}-\frac{40\!\cdots\!97}{11\!\cdots\!33}a^{5}-\frac{85\!\cdots\!19}{11\!\cdots\!33}a^{4}+\frac{27\!\cdots\!09}{22\!\cdots\!66}a^{3}-\frac{65\!\cdots\!52}{11\!\cdots\!33}a^{2}-\frac{38\!\cdots\!61}{22\!\cdots\!66}a+\frac{56\!\cdots\!09}{22\!\cdots\!66}$, $\frac{11\!\cdots\!10}{11\!\cdots\!33}a^{19}-\frac{59\!\cdots\!28}{11\!\cdots\!33}a^{18}-\frac{54\!\cdots\!98}{11\!\cdots\!33}a^{17}+\frac{78\!\cdots\!29}{11\!\cdots\!33}a^{16}+\frac{94\!\cdots\!72}{11\!\cdots\!33}a^{15}+\frac{13\!\cdots\!90}{11\!\cdots\!33}a^{14}-\frac{76\!\cdots\!02}{11\!\cdots\!33}a^{13}-\frac{26\!\cdots\!34}{11\!\cdots\!33}a^{12}+\frac{30\!\cdots\!74}{11\!\cdots\!33}a^{11}+\frac{15\!\cdots\!94}{11\!\cdots\!33}a^{10}-\frac{55\!\cdots\!68}{11\!\cdots\!33}a^{9}-\frac{35\!\cdots\!67}{11\!\cdots\!33}a^{8}+\frac{23\!\cdots\!36}{11\!\cdots\!33}a^{7}+\frac{34\!\cdots\!46}{11\!\cdots\!33}a^{6}+\frac{42\!\cdots\!08}{11\!\cdots\!33}a^{5}-\frac{80\!\cdots\!89}{11\!\cdots\!33}a^{4}-\frac{41\!\cdots\!70}{11\!\cdots\!33}a^{3}-\frac{41\!\cdots\!05}{11\!\cdots\!33}a^{2}+\frac{84\!\cdots\!69}{11\!\cdots\!33}a-\frac{81\!\cdots\!73}{11\!\cdots\!33}$, $\frac{49\!\cdots\!73}{11\!\cdots\!33}a^{19}-\frac{27\!\cdots\!85}{22\!\cdots\!66}a^{18}-\frac{48\!\cdots\!59}{22\!\cdots\!66}a^{17}-\frac{33\!\cdots\!91}{22\!\cdots\!66}a^{16}+\frac{43\!\cdots\!00}{11\!\cdots\!33}a^{15}+\frac{14\!\cdots\!73}{11\!\cdots\!33}a^{14}-\frac{76\!\cdots\!87}{22\!\cdots\!66}a^{13}-\frac{18\!\cdots\!18}{11\!\cdots\!33}a^{12}+\frac{35\!\cdots\!23}{22\!\cdots\!66}a^{11}+\frac{19\!\cdots\!13}{22\!\cdots\!66}a^{10}-\frac{44\!\cdots\!43}{11\!\cdots\!33}a^{9}-\frac{24\!\cdots\!00}{11\!\cdots\!33}a^{8}+\frac{57\!\cdots\!77}{11\!\cdots\!33}a^{7}+\frac{52\!\cdots\!77}{22\!\cdots\!66}a^{6}-\frac{36\!\cdots\!89}{11\!\cdots\!33}a^{5}-\frac{93\!\cdots\!08}{11\!\cdots\!33}a^{4}+\frac{19\!\cdots\!69}{22\!\cdots\!66}a^{3}-\frac{69\!\cdots\!57}{11\!\cdots\!33}a^{2}-\frac{21\!\cdots\!23}{22\!\cdots\!66}a+\frac{40\!\cdots\!63}{22\!\cdots\!66}$, $\frac{89\!\cdots\!21}{22\!\cdots\!66}a^{19}+\frac{31\!\cdots\!15}{22\!\cdots\!66}a^{18}-\frac{22\!\cdots\!44}{11\!\cdots\!33}a^{17}-\frac{30\!\cdots\!59}{22\!\cdots\!66}a^{16}+\frac{88\!\cdots\!13}{22\!\cdots\!66}a^{15}+\frac{39\!\cdots\!31}{11\!\cdots\!33}a^{14}-\frac{84\!\cdots\!27}{22\!\cdots\!66}a^{13}-\frac{89\!\cdots\!73}{22\!\cdots\!66}a^{12}+\frac{22\!\cdots\!02}{11\!\cdots\!33}a^{11}+\frac{25\!\cdots\!96}{11\!\cdots\!33}a^{10}-\frac{13\!\cdots\!69}{22\!\cdots\!66}a^{9}-\frac{16\!\cdots\!65}{22\!\cdots\!66}a^{8}+\frac{21\!\cdots\!47}{22\!\cdots\!66}a^{7}+\frac{28\!\cdots\!09}{22\!\cdots\!66}a^{6}-\frac{91\!\cdots\!37}{11\!\cdots\!33}a^{5}-\frac{25\!\cdots\!99}{22\!\cdots\!66}a^{4}+\frac{56\!\cdots\!87}{22\!\cdots\!66}a^{3}+\frac{46\!\cdots\!90}{11\!\cdots\!33}a^{2}+\frac{31\!\cdots\!65}{22\!\cdots\!66}a-\frac{23\!\cdots\!09}{11\!\cdots\!33}$, $\frac{91\!\cdots\!44}{11\!\cdots\!33}a^{19}+\frac{11\!\cdots\!26}{11\!\cdots\!33}a^{18}-\frac{47\!\cdots\!46}{11\!\cdots\!33}a^{17}-\frac{54\!\cdots\!82}{11\!\cdots\!33}a^{16}+\frac{15\!\cdots\!03}{22\!\cdots\!66}a^{15}+\frac{20\!\cdots\!67}{22\!\cdots\!66}a^{14}-\frac{83\!\cdots\!87}{22\!\cdots\!66}a^{13}-\frac{18\!\cdots\!43}{22\!\cdots\!66}a^{12}-\frac{14\!\cdots\!45}{22\!\cdots\!66}a^{11}+\frac{41\!\cdots\!15}{11\!\cdots\!33}a^{10}+\frac{28\!\cdots\!05}{22\!\cdots\!66}a^{9}-\frac{10\!\cdots\!69}{11\!\cdots\!33}a^{8}-\frac{94\!\cdots\!97}{22\!\cdots\!66}a^{7}+\frac{26\!\cdots\!83}{22\!\cdots\!66}a^{6}+\frac{63\!\cdots\!34}{11\!\cdots\!33}a^{5}-\frac{15\!\cdots\!15}{22\!\cdots\!66}a^{4}-\frac{32\!\cdots\!55}{11\!\cdots\!33}a^{3}+\frac{16\!\cdots\!25}{11\!\cdots\!33}a^{2}+\frac{86\!\cdots\!27}{22\!\cdots\!66}a-\frac{19\!\cdots\!81}{22\!\cdots\!66}$, $\frac{16\!\cdots\!77}{11\!\cdots\!33}a^{19}-\frac{18\!\cdots\!23}{11\!\cdots\!33}a^{18}-\frac{81\!\cdots\!73}{11\!\cdots\!33}a^{17}+\frac{58\!\cdots\!75}{11\!\cdots\!33}a^{16}+\frac{29\!\cdots\!53}{22\!\cdots\!66}a^{15}-\frac{13\!\cdots\!91}{22\!\cdots\!66}a^{14}-\frac{26\!\cdots\!97}{22\!\cdots\!66}a^{13}+\frac{74\!\cdots\!01}{22\!\cdots\!66}a^{12}+\frac{12\!\cdots\!45}{22\!\cdots\!66}a^{11}-\frac{11\!\cdots\!91}{11\!\cdots\!33}a^{10}-\frac{34\!\cdots\!29}{22\!\cdots\!66}a^{9}+\frac{21\!\cdots\!98}{11\!\cdots\!33}a^{8}+\frac{50\!\cdots\!55}{22\!\cdots\!66}a^{7}-\frac{54\!\cdots\!11}{22\!\cdots\!66}a^{6}-\frac{18\!\cdots\!25}{11\!\cdots\!33}a^{5}+\frac{39\!\cdots\!93}{22\!\cdots\!66}a^{4}+\frac{55\!\cdots\!72}{11\!\cdots\!33}a^{3}-\frac{49\!\cdots\!37}{11\!\cdots\!33}a^{2}-\frac{81\!\cdots\!17}{22\!\cdots\!66}a+\frac{13\!\cdots\!59}{22\!\cdots\!66}$, $\frac{70\!\cdots\!85}{22\!\cdots\!66}a^{19}+\frac{12\!\cdots\!35}{11\!\cdots\!33}a^{18}-\frac{34\!\cdots\!93}{22\!\cdots\!66}a^{17}-\frac{23\!\cdots\!95}{22\!\cdots\!66}a^{16}+\frac{63\!\cdots\!77}{22\!\cdots\!66}a^{15}+\frac{29\!\cdots\!00}{11\!\cdots\!33}a^{14}-\frac{27\!\cdots\!83}{11\!\cdots\!33}a^{13}-\frac{60\!\cdots\!39}{22\!\cdots\!66}a^{12}+\frac{12\!\cdots\!72}{11\!\cdots\!33}a^{11}+\frac{15\!\cdots\!81}{11\!\cdots\!33}a^{10}-\frac{31\!\cdots\!66}{11\!\cdots\!33}a^{9}-\frac{39\!\cdots\!66}{11\!\cdots\!33}a^{8}+\frac{76\!\cdots\!03}{22\!\cdots\!66}a^{7}+\frac{10\!\cdots\!59}{22\!\cdots\!66}a^{6}-\frac{40\!\cdots\!85}{22\!\cdots\!66}a^{5}-\frac{29\!\cdots\!62}{11\!\cdots\!33}a^{4}+\frac{23\!\cdots\!65}{11\!\cdots\!33}a^{3}+\frac{87\!\cdots\!55}{22\!\cdots\!66}a^{2}+\frac{55\!\cdots\!65}{11\!\cdots\!33}a+\frac{23\!\cdots\!34}{11\!\cdots\!33}$, $\frac{15\!\cdots\!27}{22\!\cdots\!66}a^{19}-\frac{59\!\cdots\!94}{11\!\cdots\!33}a^{18}-\frac{36\!\cdots\!79}{11\!\cdots\!33}a^{17}+\frac{15\!\cdots\!25}{11\!\cdots\!33}a^{16}+\frac{65\!\cdots\!21}{11\!\cdots\!33}a^{15}-\frac{21\!\cdots\!53}{22\!\cdots\!66}a^{14}-\frac{11\!\cdots\!01}{22\!\cdots\!66}a^{13}-\frac{14\!\cdots\!69}{22\!\cdots\!66}a^{12}+\frac{53\!\cdots\!19}{22\!\cdots\!66}a^{11}+\frac{29\!\cdots\!52}{11\!\cdots\!33}a^{10}-\frac{13\!\cdots\!95}{22\!\cdots\!66}a^{9}-\frac{19\!\cdots\!69}{22\!\cdots\!66}a^{8}+\frac{17\!\cdots\!71}{22\!\cdots\!66}a^{7}+\frac{11\!\cdots\!55}{11\!\cdots\!33}a^{6}-\frac{11\!\cdots\!35}{22\!\cdots\!66}a^{5}-\frac{35\!\cdots\!38}{11\!\cdots\!33}a^{4}+\frac{30\!\cdots\!95}{22\!\cdots\!66}a^{3}-\frac{12\!\cdots\!47}{11\!\cdots\!33}a^{2}-\frac{13\!\cdots\!10}{11\!\cdots\!33}a+\frac{34\!\cdots\!54}{11\!\cdots\!33}$, $\frac{68\!\cdots\!21}{11\!\cdots\!33}a^{19}+\frac{13\!\cdots\!47}{22\!\cdots\!66}a^{18}-\frac{34\!\cdots\!38}{11\!\cdots\!33}a^{17}-\frac{44\!\cdots\!70}{11\!\cdots\!33}a^{16}+\frac{12\!\cdots\!83}{22\!\cdots\!66}a^{15}+\frac{18\!\cdots\!37}{22\!\cdots\!66}a^{14}-\frac{58\!\cdots\!51}{11\!\cdots\!33}a^{13}-\frac{92\!\cdots\!43}{11\!\cdots\!33}a^{12}+\frac{27\!\cdots\!48}{11\!\cdots\!33}a^{11}+\frac{91\!\cdots\!93}{22\!\cdots\!66}a^{10}-\frac{13\!\cdots\!81}{22\!\cdots\!66}a^{9}-\frac{23\!\cdots\!01}{22\!\cdots\!66}a^{8}+\frac{83\!\cdots\!06}{11\!\cdots\!33}a^{7}+\frac{15\!\cdots\!10}{11\!\cdots\!33}a^{6}-\frac{38\!\cdots\!60}{11\!\cdots\!33}a^{5}-\frac{98\!\cdots\!13}{11\!\cdots\!33}a^{4}-\frac{25\!\cdots\!76}{11\!\cdots\!33}a^{3}+\frac{40\!\cdots\!63}{22\!\cdots\!66}a^{2}+\frac{33\!\cdots\!37}{22\!\cdots\!66}a-\frac{18\!\cdots\!53}{22\!\cdots\!66}$, $\frac{14\!\cdots\!28}{11\!\cdots\!33}a^{19}-\frac{11\!\cdots\!45}{22\!\cdots\!66}a^{18}-\frac{13\!\cdots\!37}{22\!\cdots\!66}a^{17}+\frac{48\!\cdots\!51}{22\!\cdots\!66}a^{16}+\frac{13\!\cdots\!54}{11\!\cdots\!33}a^{15}-\frac{40\!\cdots\!32}{11\!\cdots\!33}a^{14}-\frac{26\!\cdots\!45}{22\!\cdots\!66}a^{13}+\frac{33\!\cdots\!34}{11\!\cdots\!33}a^{12}+\frac{15\!\cdots\!43}{22\!\cdots\!66}a^{11}-\frac{29\!\cdots\!65}{22\!\cdots\!66}a^{10}-\frac{26\!\cdots\!83}{11\!\cdots\!33}a^{9}+\frac{35\!\cdots\!88}{11\!\cdots\!33}a^{8}+\frac{51\!\cdots\!37}{11\!\cdots\!33}a^{7}-\frac{87\!\cdots\!87}{22\!\cdots\!66}a^{6}-\frac{53\!\cdots\!29}{11\!\cdots\!33}a^{5}+\frac{24\!\cdots\!56}{11\!\cdots\!33}a^{4}+\frac{50\!\cdots\!39}{22\!\cdots\!66}a^{3}-\frac{46\!\cdots\!92}{11\!\cdots\!33}a^{2}-\frac{76\!\cdots\!65}{22\!\cdots\!66}a+\frac{67\!\cdots\!99}{22\!\cdots\!66}$, $\frac{52\!\cdots\!49}{11\!\cdots\!33}a^{19}-\frac{40\!\cdots\!11}{22\!\cdots\!66}a^{18}-\frac{51\!\cdots\!89}{22\!\cdots\!66}a^{17}+\frac{19\!\cdots\!23}{22\!\cdots\!66}a^{16}+\frac{94\!\cdots\!33}{22\!\cdots\!66}a^{15}+\frac{20\!\cdots\!01}{22\!\cdots\!66}a^{14}-\frac{42\!\cdots\!46}{11\!\cdots\!33}a^{13}-\frac{32\!\cdots\!91}{22\!\cdots\!66}a^{12}+\frac{20\!\cdots\!48}{11\!\cdots\!33}a^{11}+\frac{19\!\cdots\!35}{22\!\cdots\!66}a^{10}-\frac{10\!\cdots\!33}{22\!\cdots\!66}a^{9}-\frac{26\!\cdots\!59}{11\!\cdots\!33}a^{8}+\frac{15\!\cdots\!91}{22\!\cdots\!66}a^{7}+\frac{36\!\cdots\!01}{11\!\cdots\!33}a^{6}-\frac{54\!\cdots\!73}{11\!\cdots\!33}a^{5}-\frac{49\!\cdots\!65}{22\!\cdots\!66}a^{4}+\frac{33\!\cdots\!25}{22\!\cdots\!66}a^{3}+\frac{81\!\cdots\!23}{11\!\cdots\!33}a^{2}-\frac{20\!\cdots\!19}{11\!\cdots\!33}a-\frac{30\!\cdots\!58}{11\!\cdots\!33}$, $\frac{61\!\cdots\!31}{11\!\cdots\!33}a^{19}-\frac{17\!\cdots\!81}{22\!\cdots\!66}a^{18}+\frac{16\!\cdots\!09}{22\!\cdots\!66}a^{17}+\frac{80\!\cdots\!03}{22\!\cdots\!66}a^{16}-\frac{32\!\cdots\!93}{11\!\cdots\!33}a^{15}-\frac{70\!\cdots\!20}{11\!\cdots\!33}a^{14}+\frac{81\!\cdots\!27}{22\!\cdots\!66}a^{13}+\frac{59\!\cdots\!29}{11\!\cdots\!33}a^{12}-\frac{44\!\cdots\!17}{22\!\cdots\!66}a^{11}-\frac{53\!\cdots\!03}{22\!\cdots\!66}a^{10}+\frac{57\!\cdots\!07}{11\!\cdots\!33}a^{9}+\frac{63\!\cdots\!47}{11\!\cdots\!33}a^{8}-\frac{57\!\cdots\!16}{11\!\cdots\!33}a^{7}-\frac{15\!\cdots\!17}{22\!\cdots\!66}a^{6}-\frac{66\!\cdots\!05}{11\!\cdots\!33}a^{5}+\frac{45\!\cdots\!71}{11\!\cdots\!33}a^{4}+\frac{66\!\cdots\!25}{22\!\cdots\!66}a^{3}-\frac{10\!\cdots\!42}{11\!\cdots\!33}a^{2}-\frac{14\!\cdots\!65}{22\!\cdots\!66}a+\frac{96\!\cdots\!23}{22\!\cdots\!66}$, $\frac{65\!\cdots\!83}{11\!\cdots\!33}a^{19}-\frac{19\!\cdots\!81}{11\!\cdots\!33}a^{18}-\frac{32\!\cdots\!45}{11\!\cdots\!33}a^{17}-\frac{32\!\cdots\!41}{22\!\cdots\!66}a^{16}+\frac{62\!\cdots\!22}{11\!\cdots\!33}a^{15}+\frac{19\!\cdots\!41}{11\!\cdots\!33}a^{14}-\frac{59\!\cdots\!15}{11\!\cdots\!33}a^{13}-\frac{29\!\cdots\!55}{11\!\cdots\!33}a^{12}+\frac{61\!\cdots\!07}{22\!\cdots\!66}a^{11}+\frac{37\!\cdots\!19}{22\!\cdots\!66}a^{10}-\frac{18\!\cdots\!31}{22\!\cdots\!66}a^{9}-\frac{12\!\cdots\!95}{22\!\cdots\!66}a^{8}+\frac{14\!\cdots\!02}{11\!\cdots\!33}a^{7}+\frac{22\!\cdots\!87}{22\!\cdots\!66}a^{6}-\frac{24\!\cdots\!43}{22\!\cdots\!66}a^{5}-\frac{20\!\cdots\!69}{22\!\cdots\!66}a^{4}+\frac{40\!\cdots\!37}{11\!\cdots\!33}a^{3}+\frac{79\!\cdots\!43}{22\!\cdots\!66}a^{2}+\frac{10\!\cdots\!97}{11\!\cdots\!33}a-\frac{40\!\cdots\!71}{22\!\cdots\!66}$, $\frac{34\!\cdots\!15}{22\!\cdots\!66}a^{19}-\frac{40\!\cdots\!39}{22\!\cdots\!66}a^{18}-\frac{17\!\cdots\!15}{22\!\cdots\!66}a^{17}+\frac{14\!\cdots\!53}{22\!\cdots\!66}a^{16}+\frac{34\!\cdots\!59}{22\!\cdots\!66}a^{15}-\frac{92\!\cdots\!69}{11\!\cdots\!33}a^{14}-\frac{34\!\cdots\!93}{22\!\cdots\!66}a^{13}+\frac{55\!\cdots\!53}{11\!\cdots\!33}a^{12}+\frac{19\!\cdots\!03}{22\!\cdots\!66}a^{11}-\frac{30\!\cdots\!85}{22\!\cdots\!66}a^{10}-\frac{32\!\cdots\!01}{11\!\cdots\!33}a^{9}+\frac{18\!\cdots\!07}{22\!\cdots\!66}a^{8}+\frac{61\!\cdots\!75}{11\!\cdots\!33}a^{7}+\frac{40\!\cdots\!76}{11\!\cdots\!33}a^{6}-\frac{12\!\cdots\!03}{22\!\cdots\!66}a^{5}-\frac{17\!\cdots\!53}{22\!\cdots\!66}a^{4}+\frac{29\!\cdots\!61}{11\!\cdots\!33}a^{3}+\frac{59\!\cdots\!66}{11\!\cdots\!33}a^{2}-\frac{40\!\cdots\!70}{11\!\cdots\!33}a-\frac{98\!\cdots\!55}{11\!\cdots\!33}$, $\frac{95\!\cdots\!75}{11\!\cdots\!33}a^{19}-\frac{17\!\cdots\!15}{22\!\cdots\!66}a^{18}-\frac{91\!\cdots\!19}{22\!\cdots\!66}a^{17}+\frac{52\!\cdots\!69}{22\!\cdots\!66}a^{16}+\frac{83\!\cdots\!05}{11\!\cdots\!33}a^{15}-\frac{25\!\cdots\!72}{11\!\cdots\!33}a^{14}-\frac{15\!\cdots\!49}{22\!\cdots\!66}a^{13}+\frac{93\!\cdots\!94}{11\!\cdots\!33}a^{12}+\frac{72\!\cdots\!51}{22\!\cdots\!66}a^{11}-\frac{14\!\cdots\!95}{22\!\cdots\!66}a^{10}-\frac{95\!\cdots\!36}{11\!\cdots\!33}a^{9}-\frac{22\!\cdots\!72}{11\!\cdots\!33}a^{8}+\frac{13\!\cdots\!70}{11\!\cdots\!33}a^{7}+\frac{52\!\cdots\!17}{22\!\cdots\!66}a^{6}-\frac{94\!\cdots\!18}{11\!\cdots\!33}a^{5}+\frac{30\!\cdots\!02}{11\!\cdots\!33}a^{4}+\frac{54\!\cdots\!37}{22\!\cdots\!66}a^{3}-\frac{40\!\cdots\!31}{11\!\cdots\!33}a^{2}-\frac{49\!\cdots\!21}{22\!\cdots\!66}a+\frac{96\!\cdots\!07}{22\!\cdots\!66}$
| 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: | \( 124763101953 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 124763101953 \cdot 1}{2\cdot\sqrt{131527565972137936816816034072938673}}\cr\approx \mathstrut & 0.180363099512910 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033)
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 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033, 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 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033);
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 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033);
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
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 cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{11})^+\), 10.10.304358957700017.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | $20$ | $20$ | $20$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | $20$ | $20$ | $20$ | $20$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.20.16.1 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |
|
\(17\)
| 17.20.15.1 | $x^{20} + 89 x^{16} + 56 x^{15} + 856 x^{12} - 61712 x^{11} + 1176 x^{10} + 51344 x^{8} + 2594368 x^{7} + 1135232 x^{6} + 10976 x^{5} + 1044736 x^{4} - 10038784 x^{3} + 6165376 x^{2} - 921984 x + 1152528$ | $4$ | $5$ | $15$ | 20T1 | $[\ ]_{4}^{5}$ |