Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*x - 1)
gp: K = bnfinit(y^20 - 4*y^19 - 12*y^18 + 54*y^17 + 52*y^16 - 266*y^15 - 212*y^14 + 826*y^13 + 698*y^12 - 1923*y^11 - 1232*y^10 + 2994*y^9 + 1066*y^8 - 2717*y^7 - 451*y^6 + 1303*y^5 + 116*y^4 - 291*y^3 - 19*y^2 + 19*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*x - 1)
\( x^{20} - 4 x^{19} - 12 x^{18} + 54 x^{17} + 52 x^{16} - 266 x^{15} - 212 x^{14} + 826 x^{13} + 698 x^{12} + \cdots - 1 \)
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: | $[14, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-206548481879697235603429344256\)
\(\medspace = -\,2^{10}\cdot 61^{6}\cdot 397^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.22\) | 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: | $2^{31/16}61^{1/2}397^{1/2}\approx 596.0816979480248$ | ||
| Ramified primes: |
\(2\), \(61\), \(397\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{63495653066899}a^{19}+\frac{1027194892863}{63495653066899}a^{18}-\frac{15554134553536}{63495653066899}a^{17}+\frac{7161995983266}{63495653066899}a^{16}-\frac{22164449840582}{63495653066899}a^{15}-\frac{8683233492448}{63495653066899}a^{14}-\frac{8146728020198}{63495653066899}a^{13}-\frac{15829035979322}{63495653066899}a^{12}+\frac{4666768517859}{63495653066899}a^{11}+\frac{18161614574798}{63495653066899}a^{10}+\frac{7665412093839}{63495653066899}a^{9}-\frac{11996541288165}{63495653066899}a^{8}+\frac{13868991534572}{63495653066899}a^{7}+\frac{30514040617849}{63495653066899}a^{6}-\frac{12287065702025}{63495653066899}a^{5}+\frac{26653393855695}{63495653066899}a^{4}-\frac{20407431417955}{63495653066899}a^{3}-\frac{358603452819}{63495653066899}a^{2}-\frac{20980385250968}{63495653066899}a-\frac{4968051734960}{63495653066899}$
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: | $16$ | 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: |
$\frac{299247949213303}{63495653066899}a^{19}-\frac{13\!\cdots\!16}{63495653066899}a^{18}-\frac{28\!\cdots\!03}{63495653066899}a^{17}+\frac{17\!\cdots\!06}{63495653066899}a^{16}+\frac{57\!\cdots\!87}{63495653066899}a^{15}-\frac{81\!\cdots\!48}{63495653066899}a^{14}-\frac{17\!\cdots\!28}{63495653066899}a^{13}+\frac{25\!\cdots\!16}{63495653066899}a^{12}+\frac{66\!\cdots\!38}{63495653066899}a^{11}-\frac{59\!\cdots\!63}{63495653066899}a^{10}-\frac{25\!\cdots\!15}{63495653066899}a^{9}+\frac{88\!\cdots\!10}{63495653066899}a^{8}-\frac{19\!\cdots\!82}{63495653066899}a^{7}-\frac{67\!\cdots\!82}{63495653066899}a^{6}+\frac{25\!\cdots\!28}{63495653066899}a^{5}+\frac{22\!\cdots\!55}{63495653066899}a^{4}-\frac{97\!\cdots\!13}{63495653066899}a^{3}-\frac{26\!\cdots\!69}{63495653066899}a^{2}+\frac{10\!\cdots\!58}{63495653066899}a-\frac{486172865470064}{63495653066899}$, $\frac{244728711300423}{63495653066899}a^{19}-\frac{11\!\cdots\!87}{63495653066899}a^{18}-\frac{22\!\cdots\!61}{63495653066899}a^{17}+\frac{14\!\cdots\!89}{63495653066899}a^{16}+\frac{40\!\cdots\!50}{63495653066899}a^{15}-\frac{67\!\cdots\!01}{63495653066899}a^{14}-\frac{11\!\cdots\!68}{63495653066899}a^{13}+\frac{20\!\cdots\!87}{63495653066899}a^{12}+\frac{46\!\cdots\!27}{63495653066899}a^{11}-\frac{50\!\cdots\!21}{63495653066899}a^{10}-\frac{53\!\cdots\!07}{63495653066899}a^{9}+\frac{74\!\cdots\!76}{63495653066899}a^{8}-\frac{17\!\cdots\!54}{63495653066899}a^{7}-\frac{56\!\cdots\!13}{63495653066899}a^{6}+\frac{22\!\cdots\!65}{63495653066899}a^{5}+\frac{19\!\cdots\!85}{63495653066899}a^{4}-\frac{82\!\cdots\!60}{63495653066899}a^{3}-\frac{24\!\cdots\!49}{63495653066899}a^{2}+\frac{81\!\cdots\!40}{63495653066899}a-\frac{145253836510423}{63495653066899}$, $\frac{75708420732371}{63495653066899}a^{19}-\frac{333015622168252}{63495653066899}a^{18}-\frac{763466470127382}{63495653066899}a^{17}+\frac{43\!\cdots\!54}{63495653066899}a^{16}+\frac{20\!\cdots\!09}{63495653066899}a^{15}-\frac{20\!\cdots\!55}{63495653066899}a^{14}-\frac{68\!\cdots\!18}{63495653066899}a^{13}+\frac{62\!\cdots\!38}{63495653066899}a^{12}+\frac{23\!\cdots\!36}{63495653066899}a^{11}-\frac{14\!\cdots\!73}{63495653066899}a^{10}-\frac{20\!\cdots\!46}{63495653066899}a^{9}+\frac{22\!\cdots\!21}{63495653066899}a^{8}-\frac{33\!\cdots\!00}{63495653066899}a^{7}-\frac{17\!\cdots\!19}{63495653066899}a^{6}+\frac{60\!\cdots\!60}{63495653066899}a^{5}+\frac{62\!\cdots\!78}{63495653066899}a^{4}-\frac{28\!\cdots\!50}{63495653066899}a^{3}-\frac{72\!\cdots\!50}{63495653066899}a^{2}+\frac{40\!\cdots\!47}{63495653066899}a-\frac{177842993440111}{63495653066899}$, $\frac{180921330598261}{63495653066899}a^{19}-\frac{798929046358454}{63495653066899}a^{18}-\frac{17\!\cdots\!55}{63495653066899}a^{17}+\frac{10\!\cdots\!44}{63495653066899}a^{16}+\frac{45\!\cdots\!60}{63495653066899}a^{15}-\frac{47\!\cdots\!70}{63495653066899}a^{14}-\frac{15\!\cdots\!17}{63495653066899}a^{13}+\frac{14\!\cdots\!80}{63495653066899}a^{12}+\frac{54\!\cdots\!33}{63495653066899}a^{11}-\frac{33\!\cdots\!37}{63495653066899}a^{10}-\frac{46\!\cdots\!85}{63495653066899}a^{9}+\frac{49\!\cdots\!92}{63495653066899}a^{8}-\frac{70\!\cdots\!60}{63495653066899}a^{7}-\frac{36\!\cdots\!38}{63495653066899}a^{6}+\frac{11\!\cdots\!91}{63495653066899}a^{5}+\frac{11\!\cdots\!08}{63495653066899}a^{4}-\frac{43\!\cdots\!39}{63495653066899}a^{3}-\frac{12\!\cdots\!21}{63495653066899}a^{2}+\frac{43\!\cdots\!24}{63495653066899}a-\frac{186085766746616}{63495653066899}$, $\frac{185967152951586}{63495653066899}a^{19}-\frac{864495806059712}{63495653066899}a^{18}-\frac{16\!\cdots\!50}{63495653066899}a^{17}+\frac{11\!\cdots\!48}{63495653066899}a^{16}+\frac{26\!\cdots\!43}{63495653066899}a^{15}-\frac{52\!\cdots\!38}{63495653066899}a^{14}-\frac{71\!\cdots\!76}{63495653066899}a^{13}+\frac{16\!\cdots\!18}{63495653066899}a^{12}+\frac{30\!\cdots\!49}{63495653066899}a^{11}-\frac{38\!\cdots\!63}{63495653066899}a^{10}+\frac{46\!\cdots\!89}{63495653066899}a^{9}+\frac{57\!\cdots\!77}{63495653066899}a^{8}-\frac{14\!\cdots\!95}{63495653066899}a^{7}-\frac{44\!\cdots\!24}{63495653066899}a^{6}+\frac{18\!\cdots\!77}{63495653066899}a^{5}+\frac{14\!\cdots\!87}{63495653066899}a^{4}-\frac{67\!\cdots\!64}{63495653066899}a^{3}-\frac{17\!\cdots\!72}{63495653066899}a^{2}+\frac{70\!\cdots\!80}{63495653066899}a-\frac{262723095695261}{63495653066899}$, $\frac{478522846902738}{63495653066899}a^{19}-\frac{21\!\cdots\!13}{63495653066899}a^{18}-\frac{44\!\cdots\!35}{63495653066899}a^{17}+\frac{28\!\cdots\!55}{63495653066899}a^{16}+\frac{86\!\cdots\!68}{63495653066899}a^{15}-\frac{13\!\cdots\!40}{63495653066899}a^{14}-\frac{25\!\cdots\!33}{63495653066899}a^{13}+\frac{41\!\cdots\!83}{63495653066899}a^{12}+\frac{98\!\cdots\!62}{63495653066899}a^{11}-\frac{98\!\cdots\!27}{63495653066899}a^{10}-\frac{30\!\cdots\!55}{63495653066899}a^{9}+\frac{14\!\cdots\!69}{63495653066899}a^{8}-\frac{32\!\cdots\!74}{63495653066899}a^{7}-\frac{11\!\cdots\!04}{63495653066899}a^{6}+\frac{43\!\cdots\!21}{63495653066899}a^{5}+\frac{39\!\cdots\!63}{63495653066899}a^{4}-\frac{16\!\cdots\!17}{63495653066899}a^{3}-\frac{47\!\cdots\!86}{63495653066899}a^{2}+\frac{18\!\cdots\!77}{63495653066899}a-\frac{10\!\cdots\!64}{63495653066899}$, $\frac{497849989689880}{63495653066899}a^{19}-\frac{22\!\cdots\!79}{63495653066899}a^{18}-\frac{48\!\cdots\!03}{63495653066899}a^{17}+\frac{29\!\cdots\!13}{63495653066899}a^{16}+\frac{11\!\cdots\!54}{63495653066899}a^{15}-\frac{13\!\cdots\!36}{63495653066899}a^{14}-\frac{37\!\cdots\!12}{63495653066899}a^{13}+\frac{42\!\cdots\!63}{63495653066899}a^{12}+\frac{13\!\cdots\!09}{63495653066899}a^{11}-\frac{10\!\cdots\!92}{63495653066899}a^{10}-\frac{10\!\cdots\!22}{63495653066899}a^{9}+\frac{15\!\cdots\!80}{63495653066899}a^{8}-\frac{23\!\cdots\!84}{63495653066899}a^{7}-\frac{11\!\cdots\!59}{63495653066899}a^{6}+\frac{38\!\cdots\!75}{63495653066899}a^{5}+\frac{42\!\cdots\!00}{63495653066899}a^{4}-\frac{15\!\cdots\!97}{63495653066899}a^{3}-\frac{55\!\cdots\!21}{63495653066899}a^{2}+\frac{18\!\cdots\!60}{63495653066899}a-\frac{754648398067815}{63495653066899}$, $a$, $\frac{238643935399194}{63495653066899}a^{19}-\frac{11\!\cdots\!59}{63495653066899}a^{18}-\frac{21\!\cdots\!82}{63495653066899}a^{17}+\frac{14\!\cdots\!64}{63495653066899}a^{16}+\frac{28\!\cdots\!61}{63495653066899}a^{15}-\frac{68\!\cdots\!23}{63495653066899}a^{14}-\frac{62\!\cdots\!61}{63495653066899}a^{13}+\frac{21\!\cdots\!29}{63495653066899}a^{12}+\frac{32\!\cdots\!56}{63495653066899}a^{11}-\frac{52\!\cdots\!45}{63495653066899}a^{10}+\frac{18\!\cdots\!56}{63495653066899}a^{9}+\frac{78\!\cdots\!68}{63495653066899}a^{8}-\frac{20\!\cdots\!07}{63495653066899}a^{7}-\frac{61\!\cdots\!44}{63495653066899}a^{6}+\frac{24\!\cdots\!08}{63495653066899}a^{5}+\frac{21\!\cdots\!54}{63495653066899}a^{4}-\frac{94\!\cdots\!05}{63495653066899}a^{3}-\frac{28\!\cdots\!36}{63495653066899}a^{2}+\frac{10\!\cdots\!65}{63495653066899}a-\frac{391848268711565}{63495653066899}$, $\frac{120413636849137}{63495653066899}a^{19}-\frac{566749189830345}{63495653066899}a^{18}-\frac{10\!\cdots\!39}{63495653066899}a^{17}+\frac{73\!\cdots\!01}{63495653066899}a^{16}+\frac{13\!\cdots\!69}{63495653066899}a^{15}-\frac{34\!\cdots\!68}{63495653066899}a^{14}-\frac{28\!\cdots\!33}{63495653066899}a^{13}+\frac{10\!\cdots\!61}{63495653066899}a^{12}+\frac{14\!\cdots\!16}{63495653066899}a^{11}-\frac{25\!\cdots\!05}{63495653066899}a^{10}+\frac{14\!\cdots\!49}{63495653066899}a^{9}+\frac{38\!\cdots\!34}{63495653066899}a^{8}-\frac{11\!\cdots\!96}{63495653066899}a^{7}-\frac{29\!\cdots\!06}{63495653066899}a^{6}+\frac{12\!\cdots\!78}{63495653066899}a^{5}+\frac{10\!\cdots\!98}{63495653066899}a^{4}-\frac{47\!\cdots\!62}{63495653066899}a^{3}-\frac{13\!\cdots\!17}{63495653066899}a^{2}+\frac{48\!\cdots\!56}{63495653066899}a-\frac{180981567425533}{63495653066899}$, $a-1$, $\frac{626524297025464}{63495653066899}a^{19}-\frac{28\!\cdots\!62}{63495653066899}a^{18}-\frac{60\!\cdots\!76}{63495653066899}a^{17}+\frac{36\!\cdots\!26}{63495653066899}a^{16}+\frac{12\!\cdots\!94}{63495653066899}a^{15}-\frac{17\!\cdots\!00}{63495653066899}a^{14}-\frac{40\!\cdots\!33}{63495653066899}a^{13}+\frac{53\!\cdots\!20}{63495653066899}a^{12}+\frac{15\!\cdots\!99}{63495653066899}a^{11}-\frac{12\!\cdots\!23}{63495653066899}a^{10}-\frac{88\!\cdots\!27}{63495653066899}a^{9}+\frac{19\!\cdots\!70}{63495653066899}a^{8}-\frac{35\!\cdots\!60}{63495653066899}a^{7}-\frac{14\!\cdots\!14}{63495653066899}a^{6}+\frac{51\!\cdots\!38}{63495653066899}a^{5}+\frac{52\!\cdots\!48}{63495653066899}a^{4}-\frac{20\!\cdots\!26}{63495653066899}a^{3}-\frac{65\!\cdots\!84}{63495653066899}a^{2}+\frac{23\!\cdots\!03}{63495653066899}a-\frac{11\!\cdots\!06}{63495653066899}$, $\frac{38922336484516}{63495653066899}a^{19}-\frac{160970378495190}{63495653066899}a^{18}-\frac{440443903608614}{63495653066899}a^{17}+\frac{21\!\cdots\!57}{63495653066899}a^{16}+\frac{16\!\cdots\!56}{63495653066899}a^{15}-\frac{10\!\cdots\!59}{63495653066899}a^{14}-\frac{63\!\cdots\!20}{63495653066899}a^{13}+\frac{32\!\cdots\!15}{63495653066899}a^{12}+\frac{21\!\cdots\!89}{63495653066899}a^{11}-\frac{76\!\cdots\!90}{63495653066899}a^{10}-\frac{33\!\cdots\!64}{63495653066899}a^{9}+\frac{11\!\cdots\!58}{63495653066899}a^{8}+\frac{18\!\cdots\!99}{63495653066899}a^{7}-\frac{10\!\cdots\!31}{63495653066899}a^{6}+\frac{30\!\cdots\!74}{63495653066899}a^{5}+\frac{43\!\cdots\!93}{63495653066899}a^{4}-\frac{40\!\cdots\!42}{63495653066899}a^{3}-\frac{67\!\cdots\!54}{63495653066899}a^{2}+\frac{393023144523281}{63495653066899}a-\frac{76011037273525}{63495653066899}$, $\frac{408505707826547}{63495653066899}a^{19}-\frac{18\!\cdots\!28}{63495653066899}a^{18}-\frac{39\!\cdots\!28}{63495653066899}a^{17}+\frac{23\!\cdots\!04}{63495653066899}a^{16}+\frac{92\!\cdots\!93}{63495653066899}a^{15}-\frac{11\!\cdots\!84}{63495653066899}a^{14}-\frac{30\!\cdots\!07}{63495653066899}a^{13}+\frac{34\!\cdots\!82}{63495653066899}a^{12}+\frac{10\!\cdots\!28}{63495653066899}a^{11}-\frac{80\!\cdots\!91}{63495653066899}a^{10}-\frac{78\!\cdots\!45}{63495653066899}a^{9}+\frac{11\!\cdots\!33}{63495653066899}a^{8}-\frac{19\!\cdots\!00}{63495653066899}a^{7}-\frac{91\!\cdots\!56}{63495653066899}a^{6}+\frac{30\!\cdots\!88}{63495653066899}a^{5}+\frac{31\!\cdots\!16}{63495653066899}a^{4}-\frac{12\!\cdots\!21}{63495653066899}a^{3}-\frac{36\!\cdots\!43}{63495653066899}a^{2}+\frac{13\!\cdots\!40}{63495653066899}a-\frac{727927845448363}{63495653066899}$, $\frac{250296344485583}{63495653066899}a^{19}-\frac{11\!\cdots\!02}{63495653066899}a^{18}-\frac{24\!\cdots\!02}{63495653066899}a^{17}+\frac{14\!\cdots\!86}{63495653066899}a^{16}+\frac{57\!\cdots\!97}{63495653066899}a^{15}-\frac{68\!\cdots\!03}{63495653066899}a^{14}-\frac{19\!\cdots\!87}{63495653066899}a^{13}+\frac{20\!\cdots\!25}{63495653066899}a^{12}+\frac{68\!\cdots\!52}{63495653066899}a^{11}-\frac{49\!\cdots\!46}{63495653066899}a^{10}-\frac{51\!\cdots\!82}{63495653066899}a^{9}+\frac{73\!\cdots\!46}{63495653066899}a^{8}-\frac{11\!\cdots\!03}{63495653066899}a^{7}-\frac{57\!\cdots\!14}{63495653066899}a^{6}+\frac{18\!\cdots\!44}{63495653066899}a^{5}+\frac{20\!\cdots\!97}{63495653066899}a^{4}-\frac{70\!\cdots\!09}{63495653066899}a^{3}-\frac{25\!\cdots\!76}{63495653066899}a^{2}+\frac{70\!\cdots\!55}{63495653066899}a-\frac{299435728067271}{63495653066899}$, $\frac{89320726188690}{63495653066899}a^{19}-\frac{452806438394347}{63495653066899}a^{18}-\frac{687409511841165}{63495653066899}a^{17}+\frac{59\!\cdots\!33}{63495653066899}a^{16}-\frac{353766992278879}{63495653066899}a^{15}-\frac{27\!\cdots\!67}{63495653066899}a^{14}+\frac{41\!\cdots\!38}{63495653066899}a^{13}+\frac{90\!\cdots\!60}{63495653066899}a^{12}-\frac{59\!\cdots\!40}{63495653066899}a^{11}-\frac{22\!\cdots\!34}{63495653066899}a^{10}+\frac{44\!\cdots\!36}{63495653066899}a^{9}+\frac{33\!\cdots\!00}{63495653066899}a^{8}-\frac{12\!\cdots\!97}{63495653066899}a^{7}-\frac{26\!\cdots\!10}{63495653066899}a^{6}+\frac{13\!\cdots\!76}{63495653066899}a^{5}+\frac{98\!\cdots\!82}{63495653066899}a^{4}-\frac{48\!\cdots\!26}{63495653066899}a^{3}-\frac{14\!\cdots\!06}{63495653066899}a^{2}+\frac{48\!\cdots\!89}{63495653066899}a-\frac{62300522754458}{63495653066899}$
| 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: | \( 43604647.7992 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{3}\cdot 43604647.7992 \cdot 1}{2\cdot\sqrt{206548481879697235603429344256}}\cr\approx \mathstrut & 0.194962609036 \end{aligned}\] (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 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*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 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*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 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*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 - 12*x^18 + 54*x^17 + 52*x^16 - 266*x^15 - 212*x^14 + 826*x^13 + 698*x^12 - 1923*x^11 - 1232*x^10 + 2994*x^9 + 1066*x^8 - 2717*x^7 - 451*x^6 + 1303*x^5 + 116*x^4 - 291*x^3 - 19*x^2 + 19*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
$C_2^4.C_{3618}$ (as 20T808):
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 122880 |
| The 136 conjugacy class representatives for $C_2^4.C_{3618}$ |
| Character table for $C_2^4.C_{3618}$ |
Intermediate fields
| 10.10.14202376626313.1 |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.14 | $x^{10} + 4 x^{9} - 6 x^{8} + 176 x^{7} + 848 x^{6} + 2256 x^{5} + 1216 x^{4} - 1088 x^{3} - 5392 x^{2} - 5120 x - 3616$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.8.4.1 | $x^{8} + 9516 x^{7} + 33958096 x^{6} + 53858928086 x^{5} + 32035798457059 x^{4} + 3448323535094 x^{3} + 98379480266246 x^{2} + 1286299880976982 x + 96975348777163$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(397\)
| $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $8$ | $2$ | $4$ | $4$ |