Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*x - 1)
gp: K = bnfinit(y^20 - 6*y^19 - 2*y^18 + 68*y^17 - 53*y^16 - 288*y^15 + 114*y^14 + 1288*y^13 - 356*y^12 - 4480*y^11 + 4204*y^10 + 3960*y^9 - 6770*y^8 + 584*y^7 + 3098*y^6 - 980*y^5 - 577*y^4 + 170*y^3 + 50*y^2 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*x - 1)
\( x^{20} - 6 x^{19} - 2 x^{18} + 68 x^{17} - 53 x^{16} - 288 x^{15} + 114 x^{14} + 1288 x^{13} - 356 x^{12} + \cdots - 1 \)
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: | $[16, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(118633445252727603200000000000\)
\(\medspace = 2^{24}\cdot 5^{11}\cdot 3469^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.43\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(3469\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{2}$, $\frac{1}{344640455905006}a^{19}-\frac{68431306492963}{344640455905006}a^{18}-\frac{13762394080961}{172320227952503}a^{17}-\frac{76135473142813}{344640455905006}a^{16}+\frac{40729574922894}{172320227952503}a^{15}-\frac{38701160041422}{172320227952503}a^{14}-\frac{53427431779175}{344640455905006}a^{13}+\frac{2545168598772}{172320227952503}a^{12}+\frac{20338945835120}{172320227952503}a^{11}-\frac{31342156400343}{172320227952503}a^{10}-\frac{113594255784849}{344640455905006}a^{9}+\frac{5597954869196}{172320227952503}a^{8}+\frac{23131768693886}{172320227952503}a^{7}-\frac{2875397444330}{172320227952503}a^{6}+\frac{4194003632187}{344640455905006}a^{5}-\frac{61765168214191}{172320227952503}a^{4}-\frac{93657619135595}{344640455905006}a^{3}+\frac{7526159719609}{344640455905006}a^{2}-\frac{86385776536949}{344640455905006}a-\frac{51823930800677}{344640455905006}$
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: | $17$ | 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: |
$\frac{597769001}{4661145754}a^{19}-\frac{4058438871}{4661145754}a^{18}+\frac{565361900}{2330572877}a^{17}+\frac{21756832408}{2330572877}a^{16}-\frac{28967818756}{2330572877}a^{15}-\frac{171429983869}{4661145754}a^{14}+\frac{86315382558}{2330572877}a^{13}+\frac{824874529683}{4661145754}a^{12}-\frac{313315217671}{2330572877}a^{11}-\frac{1435009668705}{2330572877}a^{10}+\frac{1945028991699}{2330572877}a^{9}+\frac{768657920115}{2330572877}a^{8}-\frac{2516656691622}{2330572877}a^{7}+\frac{1792491978017}{4661145754}a^{6}+\frac{721475778015}{2330572877}a^{5}-\frac{894564733633}{4661145754}a^{4}-\frac{89398385777}{4661145754}a^{3}+\frac{51358662177}{4661145754}a^{2}+\frac{712531475}{2330572877}a+\frac{3387393471}{2330572877}$, $\frac{150111334174403}{172320227952503}a^{19}-\frac{13\!\cdots\!23}{344640455905006}a^{18}-\frac{12\!\cdots\!36}{172320227952503}a^{17}+\frac{15\!\cdots\!43}{344640455905006}a^{16}+\frac{66\!\cdots\!51}{344640455905006}a^{15}-\frac{33\!\cdots\!71}{172320227952503}a^{14}-\frac{31\!\cdots\!94}{172320227952503}a^{13}+\frac{12\!\cdots\!71}{172320227952503}a^{12}+\frac{23\!\cdots\!07}{344640455905006}a^{11}-\frac{85\!\cdots\!03}{344640455905006}a^{10}+\frac{60\!\cdots\!59}{172320227952503}a^{9}+\frac{94\!\cdots\!53}{344640455905006}a^{8}-\frac{58\!\cdots\!75}{344640455905006}a^{7}-\frac{13\!\cdots\!38}{172320227952503}a^{6}+\frac{13\!\cdots\!53}{172320227952503}a^{5}+\frac{77\!\cdots\!17}{172320227952503}a^{4}-\frac{32\!\cdots\!87}{344640455905006}a^{3}-\frac{16\!\cdots\!86}{172320227952503}a^{2}-\frac{12\!\cdots\!90}{172320227952503}a-\frac{138881161848195}{172320227952503}$, $\frac{142409934404205}{344640455905006}a^{19}-\frac{11\!\cdots\!47}{344640455905006}a^{18}+\frac{448959593061703}{172320227952503}a^{17}+\frac{12\!\cdots\!33}{344640455905006}a^{16}-\frac{21\!\cdots\!41}{344640455905006}a^{15}-\frac{48\!\cdots\!53}{344640455905006}a^{14}+\frac{73\!\cdots\!77}{344640455905006}a^{13}+\frac{12\!\cdots\!60}{172320227952503}a^{12}-\frac{13\!\cdots\!78}{172320227952503}a^{11}-\frac{86\!\cdots\!29}{344640455905006}a^{10}+\frac{65\!\cdots\!80}{172320227952503}a^{9}+\frac{23\!\cdots\!31}{172320227952503}a^{8}-\frac{17\!\cdots\!67}{344640455905006}a^{7}+\frac{58\!\cdots\!89}{344640455905006}a^{6}+\frac{59\!\cdots\!33}{344640455905006}a^{5}-\frac{18\!\cdots\!88}{172320227952503}a^{4}-\frac{57\!\cdots\!97}{344640455905006}a^{3}+\frac{22\!\cdots\!43}{172320227952503}a^{2}-\frac{293127494532528}{172320227952503}a-\frac{940964319206267}{344640455905006}$, $\frac{230819881731276}{172320227952503}a^{19}-\frac{28\!\cdots\!31}{344640455905006}a^{18}-\frac{454070168737029}{344640455905006}a^{17}+\frac{31\!\cdots\!73}{344640455905006}a^{16}-\frac{29\!\cdots\!85}{344640455905006}a^{15}-\frac{63\!\cdots\!41}{172320227952503}a^{14}+\frac{74\!\cdots\!13}{344640455905006}a^{13}+\frac{28\!\cdots\!99}{172320227952503}a^{12}-\frac{13\!\cdots\!72}{172320227952503}a^{11}-\frac{19\!\cdots\!25}{344640455905006}a^{10}+\frac{11\!\cdots\!61}{172320227952503}a^{9}+\frac{13\!\cdots\!65}{344640455905006}a^{8}-\frac{33\!\cdots\!11}{344640455905006}a^{7}+\frac{44\!\cdots\!09}{172320227952503}a^{6}+\frac{12\!\cdots\!53}{344640455905006}a^{5}-\frac{34\!\cdots\!65}{172320227952503}a^{4}-\frac{73\!\cdots\!25}{172320227952503}a^{3}+\frac{54\!\cdots\!65}{172320227952503}a^{2}+\frac{54\!\cdots\!25}{344640455905006}a-\frac{14\!\cdots\!90}{172320227952503}$, $\frac{220782910594539}{172320227952503}a^{19}-\frac{28\!\cdots\!07}{344640455905006}a^{18}+\frac{23653336710039}{344640455905006}a^{17}+\frac{31\!\cdots\!23}{344640455905006}a^{16}-\frac{33\!\cdots\!15}{344640455905006}a^{15}-\frac{64\!\cdots\!55}{172320227952503}a^{14}+\frac{47\!\cdots\!06}{172320227952503}a^{13}+\frac{59\!\cdots\!35}{344640455905006}a^{12}-\frac{33\!\cdots\!81}{344640455905006}a^{11}-\frac{20\!\cdots\!07}{344640455905006}a^{10}+\frac{12\!\cdots\!03}{172320227952503}a^{9}+\frac{74\!\cdots\!23}{172320227952503}a^{8}-\frac{35\!\cdots\!81}{344640455905006}a^{7}+\frac{42\!\cdots\!16}{172320227952503}a^{6}+\frac{70\!\cdots\!58}{172320227952503}a^{5}-\frac{69\!\cdots\!23}{344640455905006}a^{4}-\frac{18\!\cdots\!07}{344640455905006}a^{3}+\frac{52\!\cdots\!34}{172320227952503}a^{2}+\frac{83\!\cdots\!31}{344640455905006}a-\frac{16\!\cdots\!89}{344640455905006}$, $\frac{665818033329226}{172320227952503}a^{19}-\frac{75\!\cdots\!31}{344640455905006}a^{18}-\frac{20\!\cdots\!69}{172320227952503}a^{17}+\frac{84\!\cdots\!25}{344640455905006}a^{16}-\frac{51\!\cdots\!35}{344640455905006}a^{15}-\frac{34\!\cdots\!81}{344640455905006}a^{14}+\frac{69\!\cdots\!95}{344640455905006}a^{13}+\frac{15\!\cdots\!29}{344640455905006}a^{12}-\frac{11\!\cdots\!99}{172320227952503}a^{11}-\frac{25\!\cdots\!76}{172320227952503}a^{10}+\frac{24\!\cdots\!84}{172320227952503}a^{9}+\frac{20\!\cdots\!18}{172320227952503}a^{8}-\frac{76\!\cdots\!45}{344640455905006}a^{7}+\frac{14\!\cdots\!45}{344640455905006}a^{6}+\frac{30\!\cdots\!59}{344640455905006}a^{5}-\frac{14\!\cdots\!39}{344640455905006}a^{4}-\frac{18\!\cdots\!73}{172320227952503}a^{3}+\frac{22\!\cdots\!99}{344640455905006}a^{2}+\frac{61\!\cdots\!87}{172320227952503}a-\frac{59\!\cdots\!59}{344640455905006}$, $\frac{227157868331624}{172320227952503}a^{19}-\frac{26\!\cdots\!25}{344640455905006}a^{18}-\frac{621925449259521}{172320227952503}a^{17}+\frac{14\!\cdots\!08}{172320227952503}a^{16}-\frac{98\!\cdots\!64}{172320227952503}a^{15}-\frac{12\!\cdots\!27}{344640455905006}a^{14}+\frac{31\!\cdots\!23}{344640455905006}a^{13}+\frac{54\!\cdots\!53}{344640455905006}a^{12}-\frac{94\!\cdots\!99}{344640455905006}a^{11}-\frac{18\!\cdots\!45}{344640455905006}a^{10}+\frac{84\!\cdots\!00}{172320227952503}a^{9}+\frac{15\!\cdots\!51}{344640455905006}a^{8}-\frac{13\!\cdots\!91}{172320227952503}a^{7}+\frac{39\!\cdots\!41}{344640455905006}a^{6}+\frac{10\!\cdots\!99}{344640455905006}a^{5}-\frac{41\!\cdots\!03}{344640455905006}a^{4}-\frac{13\!\cdots\!37}{344640455905006}a^{3}+\frac{28\!\cdots\!32}{172320227952503}a^{2}+\frac{12\!\cdots\!67}{172320227952503}a-\frac{13\!\cdots\!17}{344640455905006}$, $\frac{36723755669951}{172320227952503}a^{19}-\frac{431253457701689}{344640455905006}a^{18}-\frac{158109826054923}{344640455905006}a^{17}+\frac{47\!\cdots\!97}{344640455905006}a^{16}-\frac{18\!\cdots\!78}{172320227952503}a^{15}-\frac{19\!\cdots\!09}{344640455905006}a^{14}+\frac{32\!\cdots\!32}{172320227952503}a^{13}+\frac{86\!\cdots\!71}{344640455905006}a^{12}-\frac{22\!\cdots\!67}{344640455905006}a^{11}-\frac{29\!\cdots\!15}{344640455905006}a^{10}+\frac{29\!\cdots\!61}{344640455905006}a^{9}+\frac{21\!\cdots\!63}{344640455905006}a^{8}-\frac{21\!\cdots\!55}{172320227952503}a^{7}+\frac{10\!\cdots\!75}{344640455905006}a^{6}+\frac{65\!\cdots\!21}{172320227952503}a^{5}-\frac{64\!\cdots\!27}{344640455905006}a^{4}-\frac{81\!\cdots\!19}{344640455905006}a^{3}+\frac{24\!\cdots\!87}{172320227952503}a^{2}-\frac{633510255606814}{172320227952503}a-\frac{14781225918579}{172320227952503}$, $\frac{16\!\cdots\!21}{344640455905006}a^{19}-\frac{99\!\cdots\!59}{344640455905006}a^{18}-\frac{10\!\cdots\!65}{172320227952503}a^{17}+\frac{11\!\cdots\!15}{344640455905006}a^{16}-\frac{99\!\cdots\!33}{344640455905006}a^{15}-\frac{22\!\cdots\!20}{172320227952503}a^{14}+\frac{23\!\cdots\!25}{344640455905006}a^{13}+\frac{20\!\cdots\!15}{344640455905006}a^{12}-\frac{41\!\cdots\!27}{172320227952503}a^{11}-\frac{35\!\cdots\!81}{172320227952503}a^{10}+\frac{38\!\cdots\!72}{172320227952503}a^{9}+\frac{51\!\cdots\!85}{344640455905006}a^{8}-\frac{11\!\cdots\!91}{344640455905006}a^{7}+\frac{13\!\cdots\!78}{172320227952503}a^{6}+\frac{44\!\cdots\!63}{344640455905006}a^{5}-\frac{22\!\cdots\!87}{344640455905006}a^{4}-\frac{53\!\cdots\!09}{344640455905006}a^{3}+\frac{35\!\cdots\!29}{344640455905006}a^{2}+\frac{10\!\cdots\!41}{172320227952503}a-\frac{47\!\cdots\!59}{172320227952503}$, $\frac{582762534195641}{344640455905006}a^{19}-\frac{18\!\cdots\!91}{172320227952503}a^{18}-\frac{399523931780859}{344640455905006}a^{17}+\frac{20\!\cdots\!89}{172320227952503}a^{16}-\frac{39\!\cdots\!47}{344640455905006}a^{15}-\frac{81\!\cdots\!24}{172320227952503}a^{14}+\frac{10\!\cdots\!97}{344640455905006}a^{13}+\frac{74\!\cdots\!59}{344640455905006}a^{12}-\frac{36\!\cdots\!63}{344640455905006}a^{11}-\frac{25\!\cdots\!79}{344640455905006}a^{10}+\frac{29\!\cdots\!89}{344640455905006}a^{9}+\frac{91\!\cdots\!16}{172320227952503}a^{8}-\frac{43\!\cdots\!93}{344640455905006}a^{7}+\frac{55\!\cdots\!60}{172320227952503}a^{6}+\frac{16\!\cdots\!29}{344640455905006}a^{5}-\frac{87\!\cdots\!27}{344640455905006}a^{4}-\frac{10\!\cdots\!43}{172320227952503}a^{3}+\frac{13\!\cdots\!25}{344640455905006}a^{2}+\frac{44\!\cdots\!22}{172320227952503}a-\frac{17\!\cdots\!83}{172320227952503}$, $\frac{615515670922243}{344640455905006}a^{19}-\frac{39\!\cdots\!15}{344640455905006}a^{18}+\frac{13227766761730}{172320227952503}a^{17}+\frac{43\!\cdots\!15}{344640455905006}a^{16}-\frac{23\!\cdots\!93}{172320227952503}a^{15}-\frac{17\!\cdots\!59}{344640455905006}a^{14}+\frac{64\!\cdots\!24}{172320227952503}a^{13}+\frac{40\!\cdots\!88}{172320227952503}a^{12}-\frac{45\!\cdots\!47}{344640455905006}a^{11}-\frac{14\!\cdots\!12}{172320227952503}a^{10}+\frac{34\!\cdots\!09}{344640455905006}a^{9}+\frac{19\!\cdots\!41}{344640455905006}a^{8}-\frac{24\!\cdots\!98}{172320227952503}a^{7}+\frac{13\!\cdots\!11}{344640455905006}a^{6}+\frac{92\!\cdots\!89}{172320227952503}a^{5}-\frac{49\!\cdots\!77}{172320227952503}a^{4}-\frac{10\!\cdots\!37}{172320227952503}a^{3}+\frac{14\!\cdots\!99}{344640455905006}a^{2}+\frac{61\!\cdots\!55}{344640455905006}a-\frac{19\!\cdots\!81}{172320227952503}$, $\frac{50680061327543}{172320227952503}a^{19}-\frac{552095462443587}{344640455905006}a^{18}-\frac{203178775211488}{172320227952503}a^{17}+\frac{31\!\cdots\!29}{172320227952503}a^{16}-\frac{28\!\cdots\!83}{344640455905006}a^{15}-\frac{12\!\cdots\!87}{172320227952503}a^{14}+\frac{773496748838159}{344640455905006}a^{13}+\frac{54\!\cdots\!32}{172320227952503}a^{12}-\frac{11\!\cdots\!85}{344640455905006}a^{11}-\frac{37\!\cdots\!95}{344640455905006}a^{10}+\frac{31\!\cdots\!47}{344640455905006}a^{9}+\frac{31\!\cdots\!25}{344640455905006}a^{8}-\frac{51\!\cdots\!85}{344640455905006}a^{7}+\frac{37\!\cdots\!37}{172320227952503}a^{6}+\frac{20\!\cdots\!29}{344640455905006}a^{5}-\frac{44\!\cdots\!86}{172320227952503}a^{4}-\frac{20\!\cdots\!75}{344640455905006}a^{3}+\frac{62\!\cdots\!92}{172320227952503}a^{2}-\frac{723194078616945}{344640455905006}a-\frac{377950534241213}{344640455905006}$, $\frac{640523308205999}{344640455905006}a^{19}-\frac{18\!\cdots\!28}{172320227952503}a^{18}-\frac{15\!\cdots\!31}{344640455905006}a^{17}+\frac{41\!\cdots\!33}{344640455905006}a^{16}-\frac{15\!\cdots\!01}{172320227952503}a^{15}-\frac{16\!\cdots\!51}{344640455905006}a^{14}+\frac{29\!\cdots\!30}{172320227952503}a^{13}+\frac{74\!\cdots\!11}{344640455905006}a^{12}-\frac{10\!\cdots\!96}{172320227952503}a^{11}-\frac{25\!\cdots\!99}{344640455905006}a^{10}+\frac{13\!\cdots\!01}{172320227952503}a^{9}+\frac{19\!\cdots\!81}{344640455905006}a^{8}-\frac{20\!\cdots\!78}{172320227952503}a^{7}+\frac{95\!\cdots\!45}{344640455905006}a^{6}+\frac{78\!\cdots\!47}{172320227952503}a^{5}-\frac{81\!\cdots\!83}{344640455905006}a^{4}-\frac{17\!\cdots\!37}{344640455905006}a^{3}+\frac{13\!\cdots\!57}{344640455905006}a^{2}+\frac{48\!\cdots\!49}{344640455905006}a-\frac{17\!\cdots\!89}{172320227952503}$, $\frac{745540938767050}{172320227952503}a^{19}-\frac{43\!\cdots\!13}{172320227952503}a^{18}-\frac{37\!\cdots\!85}{344640455905006}a^{17}+\frac{48\!\cdots\!63}{172320227952503}a^{16}-\frac{34\!\cdots\!26}{172320227952503}a^{15}-\frac{19\!\cdots\!03}{172320227952503}a^{14}+\frac{61\!\cdots\!19}{172320227952503}a^{13}+\frac{17\!\cdots\!61}{344640455905006}a^{12}-\frac{21\!\cdots\!39}{172320227952503}a^{11}-\frac{59\!\cdots\!21}{344640455905006}a^{10}+\frac{29\!\cdots\!45}{172320227952503}a^{9}+\frac{22\!\cdots\!93}{172320227952503}a^{8}-\frac{45\!\cdots\!89}{172320227952503}a^{7}+\frac{10\!\cdots\!97}{172320227952503}a^{6}+\frac{17\!\cdots\!06}{172320227952503}a^{5}-\frac{17\!\cdots\!23}{344640455905006}a^{4}-\frac{18\!\cdots\!11}{172320227952503}a^{3}+\frac{27\!\cdots\!45}{344640455905006}a^{2}+\frac{76\!\cdots\!91}{344640455905006}a-\frac{32\!\cdots\!10}{172320227952503}$, $\frac{147326887087963}{344640455905006}a^{19}-\frac{297455453731114}{172320227952503}a^{18}-\frac{14\!\cdots\!37}{344640455905006}a^{17}+\frac{34\!\cdots\!16}{172320227952503}a^{16}+\frac{28\!\cdots\!68}{172320227952503}a^{15}-\frac{14\!\cdots\!31}{172320227952503}a^{14}-\frac{20\!\cdots\!18}{172320227952503}a^{13}+\frac{97\!\cdots\!35}{344640455905006}a^{12}+\frac{14\!\cdots\!57}{344640455905006}a^{11}-\frac{16\!\cdots\!13}{172320227952503}a^{10}-\frac{22\!\cdots\!05}{344640455905006}a^{9}+\frac{33\!\cdots\!31}{344640455905006}a^{8}-\frac{85\!\cdots\!41}{172320227952503}a^{7}-\frac{31\!\cdots\!32}{172320227952503}a^{6}+\frac{31\!\cdots\!69}{172320227952503}a^{5}-\frac{15\!\cdots\!99}{344640455905006}a^{4}+\frac{24\!\cdots\!15}{172320227952503}a^{3}+\frac{34\!\cdots\!76}{172320227952503}a^{2}-\frac{10\!\cdots\!68}{172320227952503}a-\frac{747017126959397}{344640455905006}$, $\frac{154662821984265}{172320227952503}a^{19}-\frac{23\!\cdots\!71}{344640455905006}a^{18}+\frac{738681724818446}{172320227952503}a^{17}+\frac{12\!\cdots\!82}{172320227952503}a^{16}-\frac{40\!\cdots\!13}{344640455905006}a^{15}-\frac{49\!\cdots\!09}{172320227952503}a^{14}+\frac{68\!\cdots\!70}{172320227952503}a^{13}+\frac{48\!\cdots\!69}{344640455905006}a^{12}-\frac{24\!\cdots\!81}{172320227952503}a^{11}-\frac{85\!\cdots\!51}{172320227952503}a^{10}+\frac{26\!\cdots\!85}{344640455905006}a^{9}+\frac{91\!\cdots\!71}{344640455905006}a^{8}-\frac{34\!\cdots\!83}{344640455905006}a^{7}+\frac{61\!\cdots\!45}{172320227952503}a^{6}+\frac{60\!\cdots\!70}{172320227952503}a^{5}-\frac{77\!\cdots\!83}{344640455905006}a^{4}-\frac{62\!\cdots\!65}{172320227952503}a^{3}+\frac{10\!\cdots\!07}{344640455905006}a^{2}+\frac{44\!\cdots\!85}{344640455905006}a-\frac{20\!\cdots\!89}{344640455905006}$, $\frac{55121788456885}{344640455905006}a^{19}-\frac{123110187435055}{172320227952503}a^{18}-\frac{404412440832939}{344640455905006}a^{17}+\frac{27\!\cdots\!79}{344640455905006}a^{16}+\frac{375679521793610}{172320227952503}a^{15}-\frac{49\!\cdots\!06}{172320227952503}a^{14}-\frac{95\!\cdots\!57}{344640455905006}a^{13}+\frac{36\!\cdots\!19}{344640455905006}a^{12}+\frac{29\!\cdots\!59}{344640455905006}a^{11}-\frac{61\!\cdots\!36}{172320227952503}a^{10}+\frac{35\!\cdots\!43}{172320227952503}a^{9}+\frac{56\!\cdots\!51}{344640455905006}a^{8}-\frac{59\!\cdots\!75}{172320227952503}a^{7}+\frac{43\!\cdots\!04}{172320227952503}a^{6}-\frac{63\!\cdots\!93}{344640455905006}a^{5}-\frac{42\!\cdots\!93}{344640455905006}a^{4}+\frac{11\!\cdots\!54}{172320227952503}a^{3}+\frac{29\!\cdots\!67}{172320227952503}a^{2}-\frac{30\!\cdots\!53}{344640455905006}a-\frac{352892595874046}{172320227952503}$
| 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: | \( 63369027.0403 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{2}\cdot 63369027.0403 \cdot 1}{2\cdot\sqrt{118633445252727603200000000000}}\cr\approx \mathstrut & 0.238003387962 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*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 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*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 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*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 - 6*x^19 - 2*x^18 + 68*x^17 - 53*x^16 - 288*x^15 + 114*x^14 + 1288*x^13 - 356*x^12 - 4480*x^11 + 4204*x^10 + 3960*x^9 - 6770*x^8 + 584*x^7 + 3098*x^6 - 980*x^5 - 577*x^4 + 170*x^3 + 50*x^2 - 4*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
$C_2^9.D_5\wr C_2$ (as 20T755):
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 102400 |
| The 130 conjugacy class representatives for $C_2^9.D_5\wr C_2$ |
| Character table for $C_2^9.D_5\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.9627168800000.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | R | $20$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $20$ | $10$ | $2$ | $24$ | |||
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(3469\)
| $\Q_{3469}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3469}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |