Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101)
gp: K = bnfinit(y^20 - 10*y^19 + 63*y^18 - 282*y^17 + 1160*y^16 - 4180*y^15 + 14148*y^14 - 41798*y^13 + 119190*y^12 - 304054*y^11 + 755996*y^10 - 1649418*y^9 + 3590458*y^8 - 6710840*y^7 + 12787473*y^6 - 19605652*y^5 + 32733776*y^4 - 38415016*y^3 + 55028485*y^2 - 38299500*y + 51338101, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101)
\( x^{20} - 10 x^{19} + 63 x^{18} - 282 x^{17} + 1160 x^{16} - 4180 x^{15} + 14148 x^{14} - 41798 x^{13} + \cdots + 51338101 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3361869388230684433628866560000000000\)
\(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(67.04\) | 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: | $2\cdot 3^{1/2}5^{1/2}11^{9/10}\approx 67.03923376773275$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(660=2^{2}\cdot 3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(389,·)$, $\chi_{660}(391,·)$, $\chi_{660}(269,·)$, $\chi_{660}(271,·)$, $\chi_{660}(659,·)$, $\chi_{660}(151,·)$, $\chi_{660}(89,·)$, $\chi_{660}(479,·)$, $\chi_{660}(421,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(239,·)$, $\chi_{660}(449,·)$, $\chi_{660}(211,·)$, $\chi_{660}(181,·)$, $\chi_{660}(571,·)$, $\chi_{660}(509,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{34\!\cdots\!41}a^{18}-\frac{9}{34\!\cdots\!41}a^{17}-\frac{22\!\cdots\!43}{34\!\cdots\!41}a^{16}-\frac{16\!\cdots\!93}{34\!\cdots\!41}a^{15}+\frac{96\!\cdots\!43}{34\!\cdots\!41}a^{14}+\frac{37\!\cdots\!46}{34\!\cdots\!41}a^{13}-\frac{16\!\cdots\!78}{34\!\cdots\!41}a^{12}-\frac{10\!\cdots\!69}{34\!\cdots\!41}a^{11}-\frac{13\!\cdots\!55}{34\!\cdots\!41}a^{10}+\frac{92\!\cdots\!98}{34\!\cdots\!41}a^{9}+\frac{26\!\cdots\!73}{34\!\cdots\!41}a^{8}+\frac{11\!\cdots\!48}{34\!\cdots\!41}a^{7}+\frac{11\!\cdots\!86}{34\!\cdots\!41}a^{6}+\frac{39\!\cdots\!57}{34\!\cdots\!41}a^{5}+\frac{14\!\cdots\!35}{34\!\cdots\!41}a^{4}-\frac{73\!\cdots\!87}{34\!\cdots\!41}a^{3}-\frac{32\!\cdots\!76}{34\!\cdots\!41}a^{2}+\frac{29\!\cdots\!23}{34\!\cdots\!41}a+\frac{14\!\cdots\!40}{34\!\cdots\!41}$, $\frac{1}{23\!\cdots\!99}a^{19}+\frac{3347060}{23\!\cdots\!99}a^{18}+\frac{22\!\cdots\!39}{23\!\cdots\!99}a^{17}-\frac{86\!\cdots\!45}{23\!\cdots\!99}a^{16}-\frac{73\!\cdots\!06}{23\!\cdots\!99}a^{15}-\frac{61\!\cdots\!49}{23\!\cdots\!99}a^{14}+\frac{33\!\cdots\!42}{23\!\cdots\!99}a^{13}+\frac{46\!\cdots\!86}{23\!\cdots\!99}a^{12}-\frac{45\!\cdots\!61}{23\!\cdots\!99}a^{11}+\frac{32\!\cdots\!77}{23\!\cdots\!99}a^{10}-\frac{59\!\cdots\!37}{23\!\cdots\!99}a^{9}+\frac{60\!\cdots\!83}{23\!\cdots\!99}a^{8}-\frac{96\!\cdots\!94}{23\!\cdots\!99}a^{7}-\frac{24\!\cdots\!27}{23\!\cdots\!99}a^{6}-\frac{74\!\cdots\!65}{23\!\cdots\!99}a^{5}-\frac{54\!\cdots\!30}{23\!\cdots\!99}a^{4}-\frac{10\!\cdots\!58}{23\!\cdots\!99}a^{3}-\frac{14\!\cdots\!58}{23\!\cdots\!99}a^{2}-\frac{69\!\cdots\!94}{23\!\cdots\!99}a+\frac{76\!\cdots\!49}{23\!\cdots\!99}$
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
$C_{2}\times C_{7964}$, which has order $15928$ (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: | $9$ | 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: |
$\frac{12\!\cdots\!62}{34\!\cdots\!41}a^{18}-\frac{10\!\cdots\!58}{34\!\cdots\!41}a^{17}+\frac{67\!\cdots\!90}{34\!\cdots\!41}a^{16}-\frac{29\!\cdots\!72}{34\!\cdots\!41}a^{15}+\frac{12\!\cdots\!02}{34\!\cdots\!41}a^{14}-\frac{43\!\cdots\!22}{34\!\cdots\!41}a^{13}+\frac{14\!\cdots\!35}{34\!\cdots\!41}a^{12}-\frac{43\!\cdots\!76}{34\!\cdots\!41}a^{11}+\frac{12\!\cdots\!69}{34\!\cdots\!41}a^{10}-\frac{32\!\cdots\!42}{34\!\cdots\!41}a^{9}+\frac{83\!\cdots\!52}{34\!\cdots\!41}a^{8}-\frac{17\!\cdots\!60}{34\!\cdots\!41}a^{7}+\frac{42\!\cdots\!87}{34\!\cdots\!41}a^{6}-\frac{76\!\cdots\!92}{34\!\cdots\!41}a^{5}+\frac{17\!\cdots\!79}{34\!\cdots\!41}a^{4}-\frac{24\!\cdots\!94}{34\!\cdots\!41}a^{3}+\frac{51\!\cdots\!30}{34\!\cdots\!41}a^{2}-\frac{39\!\cdots\!90}{34\!\cdots\!41}a+\frac{12\!\cdots\!86}{34\!\cdots\!41}$, $\frac{11\!\cdots\!88}{34\!\cdots\!41}a^{18}-\frac{10\!\cdots\!92}{34\!\cdots\!41}a^{17}+\frac{36\!\cdots\!42}{34\!\cdots\!41}a^{16}-\frac{62\!\cdots\!84}{34\!\cdots\!41}a^{15}-\frac{23\!\cdots\!84}{34\!\cdots\!41}a^{14}+\frac{46\!\cdots\!76}{34\!\cdots\!41}a^{13}-\frac{32\!\cdots\!60}{34\!\cdots\!41}a^{12}+\frac{13\!\cdots\!76}{34\!\cdots\!41}a^{11}-\frac{48\!\cdots\!68}{34\!\cdots\!41}a^{10}+\frac{12\!\cdots\!64}{34\!\cdots\!41}a^{9}-\frac{35\!\cdots\!71}{34\!\cdots\!41}a^{8}+\frac{77\!\cdots\!00}{34\!\cdots\!41}a^{7}-\frac{17\!\cdots\!62}{34\!\cdots\!41}a^{6}+\frac{29\!\cdots\!96}{34\!\cdots\!41}a^{5}-\frac{55\!\cdots\!23}{34\!\cdots\!41}a^{4}+\frac{70\!\cdots\!92}{34\!\cdots\!41}a^{3}-\frac{11\!\cdots\!90}{34\!\cdots\!41}a^{2}+\frac{81\!\cdots\!00}{34\!\cdots\!41}a-\frac{12\!\cdots\!39}{34\!\cdots\!41}$, $\frac{12\!\cdots\!50}{34\!\cdots\!41}a^{18}-\frac{11\!\cdots\!50}{34\!\cdots\!41}a^{17}+\frac{43\!\cdots\!32}{34\!\cdots\!41}a^{16}-\frac{92\!\cdots\!56}{34\!\cdots\!41}a^{15}+\frac{99\!\cdots\!18}{34\!\cdots\!41}a^{14}+\frac{34\!\cdots\!54}{34\!\cdots\!41}a^{13}-\frac{17\!\cdots\!25}{34\!\cdots\!41}a^{12}+\frac{92\!\cdots\!00}{34\!\cdots\!41}a^{11}-\frac{36\!\cdots\!99}{34\!\cdots\!41}a^{10}+\frac{97\!\cdots\!22}{34\!\cdots\!41}a^{9}-\frac{27\!\cdots\!19}{34\!\cdots\!41}a^{8}+\frac{59\!\cdots\!40}{34\!\cdots\!41}a^{7}-\frac{13\!\cdots\!75}{34\!\cdots\!41}a^{6}+\frac{21\!\cdots\!04}{34\!\cdots\!41}a^{5}-\frac{37\!\cdots\!44}{34\!\cdots\!41}a^{4}+\frac{45\!\cdots\!98}{34\!\cdots\!41}a^{3}-\frac{60\!\cdots\!60}{34\!\cdots\!41}a^{2}+\frac{41\!\cdots\!10}{34\!\cdots\!41}a-\frac{32\!\cdots\!94}{34\!\cdots\!41}$, $\frac{96\!\cdots\!98}{34\!\cdots\!41}a^{18}-\frac{87\!\cdots\!82}{34\!\cdots\!41}a^{17}+\frac{53\!\cdots\!82}{34\!\cdots\!41}a^{16}-\frac{22\!\cdots\!64}{34\!\cdots\!41}a^{15}+\frac{94\!\cdots\!60}{34\!\cdots\!41}a^{14}-\frac{33\!\cdots\!72}{34\!\cdots\!41}a^{13}+\frac{11\!\cdots\!55}{34\!\cdots\!41}a^{12}-\frac{32\!\cdots\!10}{34\!\cdots\!41}a^{11}+\frac{94\!\cdots\!75}{34\!\cdots\!41}a^{10}-\frac{23\!\cdots\!70}{34\!\cdots\!41}a^{9}+\frac{58\!\cdots\!70}{34\!\cdots\!41}a^{8}-\frac{12\!\cdots\!62}{34\!\cdots\!41}a^{7}+\frac{27\!\cdots\!67}{34\!\cdots\!41}a^{6}-\frac{45\!\cdots\!14}{34\!\cdots\!41}a^{5}+\frac{90\!\cdots\!04}{34\!\cdots\!41}a^{4}-\frac{11\!\cdots\!96}{34\!\cdots\!41}a^{3}+\frac{19\!\cdots\!13}{34\!\cdots\!41}a^{2}-\frac{14\!\cdots\!54}{34\!\cdots\!41}a+\frac{26\!\cdots\!35}{34\!\cdots\!41}$, $\frac{13\!\cdots\!70}{23\!\cdots\!99}a^{19}-\frac{10\!\cdots\!65}{23\!\cdots\!99}a^{18}+\frac{51\!\cdots\!45}{23\!\cdots\!99}a^{17}-\frac{19\!\cdots\!50}{23\!\cdots\!99}a^{16}+\frac{74\!\cdots\!20}{23\!\cdots\!99}a^{15}-\frac{23\!\cdots\!80}{23\!\cdots\!99}a^{14}+\frac{72\!\cdots\!02}{23\!\cdots\!99}a^{13}-\frac{18\!\cdots\!23}{23\!\cdots\!99}a^{12}+\frac{50\!\cdots\!48}{23\!\cdots\!99}a^{11}-\frac{10\!\cdots\!92}{23\!\cdots\!99}a^{10}+\frac{24\!\cdots\!91}{23\!\cdots\!99}a^{9}-\frac{44\!\cdots\!59}{23\!\cdots\!99}a^{8}+\frac{92\!\cdots\!56}{23\!\cdots\!99}a^{7}-\frac{12\!\cdots\!80}{23\!\cdots\!99}a^{6}+\frac{23\!\cdots\!65}{23\!\cdots\!99}a^{5}-\frac{20\!\cdots\!28}{23\!\cdots\!99}a^{4}+\frac{39\!\cdots\!27}{23\!\cdots\!99}a^{3}-\frac{65\!\cdots\!78}{23\!\cdots\!99}a^{2}+\frac{37\!\cdots\!69}{23\!\cdots\!99}a+\frac{32\!\cdots\!74}{23\!\cdots\!99}$, $\frac{20\!\cdots\!70}{23\!\cdots\!99}a^{19}-\frac{21\!\cdots\!25}{23\!\cdots\!99}a^{18}+\frac{13\!\cdots\!95}{23\!\cdots\!99}a^{17}-\frac{59\!\cdots\!77}{23\!\cdots\!99}a^{16}+\frac{22\!\cdots\!36}{23\!\cdots\!99}a^{15}-\frac{76\!\cdots\!62}{23\!\cdots\!99}a^{14}+\frac{24\!\cdots\!76}{23\!\cdots\!99}a^{13}-\frac{67\!\cdots\!23}{23\!\cdots\!99}a^{12}+\frac{17\!\cdots\!54}{23\!\cdots\!99}a^{11}-\frac{41\!\cdots\!04}{23\!\cdots\!99}a^{10}+\frac{92\!\cdots\!63}{23\!\cdots\!99}a^{9}-\frac{18\!\cdots\!41}{23\!\cdots\!99}a^{8}+\frac{34\!\cdots\!28}{23\!\cdots\!99}a^{7}-\frac{55\!\cdots\!29}{23\!\cdots\!99}a^{6}+\frac{88\!\cdots\!20}{23\!\cdots\!99}a^{5}-\frac{98\!\cdots\!51}{23\!\cdots\!99}a^{4}+\frac{13\!\cdots\!22}{23\!\cdots\!99}a^{3}-\frac{61\!\cdots\!76}{23\!\cdots\!99}a^{2}+\frac{68\!\cdots\!78}{23\!\cdots\!99}a+\frac{12\!\cdots\!50}{23\!\cdots\!99}$, $\frac{66\!\cdots\!12}{23\!\cdots\!99}a^{19}-\frac{12\!\cdots\!36}{23\!\cdots\!99}a^{18}+\frac{97\!\cdots\!56}{23\!\cdots\!99}a^{17}-\frac{52\!\cdots\!84}{23\!\cdots\!99}a^{16}+\frac{22\!\cdots\!28}{23\!\cdots\!99}a^{15}-\frac{88\!\cdots\!38}{23\!\cdots\!99}a^{14}+\frac{30\!\cdots\!24}{23\!\cdots\!99}a^{13}-\frac{10\!\cdots\!42}{23\!\cdots\!99}a^{12}+\frac{28\!\cdots\!04}{23\!\cdots\!99}a^{11}-\frac{81\!\cdots\!48}{23\!\cdots\!99}a^{10}+\frac{20\!\cdots\!16}{23\!\cdots\!99}a^{9}-\frac{49\!\cdots\!22}{23\!\cdots\!99}a^{8}+\frac{10\!\cdots\!32}{23\!\cdots\!99}a^{7}-\frac{22\!\cdots\!06}{23\!\cdots\!99}a^{6}+\frac{38\!\cdots\!72}{23\!\cdots\!99}a^{5}-\frac{72\!\cdots\!61}{23\!\cdots\!99}a^{4}+\frac{10\!\cdots\!44}{23\!\cdots\!99}a^{3}-\frac{15\!\cdots\!32}{23\!\cdots\!99}a^{2}+\frac{12\!\cdots\!49}{23\!\cdots\!99}a-\frac{16\!\cdots\!50}{23\!\cdots\!99}$, $\frac{13\!\cdots\!70}{23\!\cdots\!99}a^{19}-\frac{12\!\cdots\!93}{23\!\cdots\!99}a^{18}+\frac{77\!\cdots\!97}{23\!\cdots\!99}a^{17}-\frac{31\!\cdots\!25}{23\!\cdots\!99}a^{16}+\frac{11\!\cdots\!08}{23\!\cdots\!99}a^{15}-\frac{35\!\cdots\!92}{23\!\cdots\!99}a^{14}+\frac{10\!\cdots\!38}{23\!\cdots\!99}a^{13}-\frac{27\!\cdots\!33}{23\!\cdots\!99}a^{12}+\frac{66\!\cdots\!88}{23\!\cdots\!99}a^{11}-\frac{13\!\cdots\!20}{23\!\cdots\!99}a^{10}+\frac{28\!\cdots\!23}{23\!\cdots\!99}a^{9}-\frac{42\!\cdots\!63}{23\!\cdots\!99}a^{8}+\frac{75\!\cdots\!88}{23\!\cdots\!99}a^{7}-\frac{54\!\cdots\!02}{23\!\cdots\!99}a^{6}+\frac{91\!\cdots\!17}{23\!\cdots\!99}a^{5}+\frac{15\!\cdots\!09}{23\!\cdots\!99}a^{4}-\frac{10\!\cdots\!89}{23\!\cdots\!99}a^{3}+\frac{87\!\cdots\!64}{23\!\cdots\!99}a^{2}-\frac{34\!\cdots\!47}{23\!\cdots\!99}a+\frac{11\!\cdots\!14}{23\!\cdots\!99}$, $\frac{26\!\cdots\!92}{23\!\cdots\!99}a^{19}-\frac{17\!\cdots\!92}{23\!\cdots\!99}a^{18}+\frac{85\!\cdots\!84}{23\!\cdots\!99}a^{17}-\frac{42\!\cdots\!62}{23\!\cdots\!99}a^{16}+\frac{22\!\cdots\!10}{23\!\cdots\!99}a^{15}-\frac{97\!\cdots\!19}{23\!\cdots\!99}a^{14}+\frac{35\!\cdots\!15}{23\!\cdots\!99}a^{13}-\frac{11\!\cdots\!78}{23\!\cdots\!99}a^{12}+\frac{35\!\cdots\!35}{23\!\cdots\!99}a^{11}-\frac{98\!\cdots\!09}{23\!\cdots\!99}a^{10}+\frac{24\!\cdots\!20}{23\!\cdots\!99}a^{9}-\frac{58\!\cdots\!92}{23\!\cdots\!99}a^{8}+\frac{12\!\cdots\!61}{23\!\cdots\!99}a^{7}-\frac{24\!\cdots\!76}{23\!\cdots\!99}a^{6}+\frac{43\!\cdots\!31}{23\!\cdots\!99}a^{5}-\frac{70\!\cdots\!91}{23\!\cdots\!99}a^{4}+\frac{98\!\cdots\!34}{23\!\cdots\!99}a^{3}-\frac{12\!\cdots\!87}{23\!\cdots\!99}a^{2}+\frac{10\!\cdots\!08}{23\!\cdots\!99}a-\frac{99\!\cdots\!63}{23\!\cdots\!99}$
| 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: | \( 281202.490766 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 281202.490766 \cdot 15928}{2\cdot\sqrt{3361869388230684433628866560000000000}}\cr\approx \mathstrut & 0.117127447694 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101)
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 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101, 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 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101);
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 - 10*x^19 + 63*x^18 - 282*x^17 + 1160*x^16 - 4180*x^15 + 14148*x^14 - 41798*x^13 + 119190*x^12 - 304054*x^11 + 755996*x^10 - 1649418*x^9 + 3590458*x^8 - 6710840*x^7 + 12787473*x^6 - 19605652*x^5 + 32733776*x^4 - 38415016*x^3 + 55028485*x^2 - 38299500*x + 51338101);
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
$C_2\times C_{10}$ (as 20T3):
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)
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{11}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 10.0.1833540124521600000.1, \(\Q(\zeta_{44})^+\), 10.0.162778775259375.1 |
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 | R | R | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.11 | $x^{10} + 10 x^{9} + 38 x^{8} + 64 x^{7} + 152 x^{6} + 688 x^{5} + 912 x^{4} - 1024 x^{3} - 1968 x^{2} + 32 x - 32$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.11 | $x^{10} + 10 x^{9} + 38 x^{8} + 64 x^{7} + 152 x^{6} + 688 x^{5} + 912 x^{4} - 1024 x^{3} - 1968 x^{2} + 32 x - 32$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(5\)
| 5.10.5.2 | $x^{10} + 2500 x^{2} - 9375$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.2 | $x^{10} + 2500 x^{2} - 9375$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |