Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113)
gp: K = bnfinit(y^20 - y^19 + y^18 - 93*y^17 + 209*y^16 - 800*y^15 + 4981*y^14 - 12556*y^13 + 52672*y^12 - 186410*y^11 + 461402*y^10 - 1486634*y^9 + 3730667*y^8 - 7453323*y^7 + 16223240*y^6 - 23433182*y^5 + 16658019*y^4 - 7158636*y^3 + 5906450*y^2 - 2576267*y + 1575113, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113)
\( x^{20} - x^{19} + x^{18} - 93 x^{17} + 209 x^{16} - 800 x^{15} + 4981 x^{14} - 12556 x^{13} + \cdots + 1575113 \)
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: |
\(396508891893913150393870750167904820789\)
\(\medspace = 11^{16}\cdot 29^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(85.10\) | 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: | $11^{4/5}29^{3/4}\approx 85.09668585773373$ | ||
| Ramified primes: |
\(11\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(319=11\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(133,·)$, $\chi_{319}(262,·)$, $\chi_{319}(202,·)$, $\chi_{319}(75,·)$, $\chi_{319}(12,·)$, $\chi_{319}(144,·)$, $\chi_{319}(273,·)$, $\chi_{319}(146,·)$, $\chi_{319}(278,·)$, $\chi_{319}(86,·)$, $\chi_{319}(157,·)$, $\chi_{319}(289,·)$, $\chi_{319}(291,·)$, $\chi_{319}(70,·)$, $\chi_{319}(104,·)$, $\chi_{319}(302,·)$, $\chi_{319}(115,·)$, $\chi_{319}(59,·)$, $\chi_{319}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23}a^{14}-\frac{7}{23}a^{13}+\frac{4}{23}a^{12}-\frac{5}{23}a^{11}-\frac{7}{23}a^{10}-\frac{7}{23}a^{9}+\frac{8}{23}a^{8}+\frac{1}{23}a^{7}-\frac{9}{23}a^{6}-\frac{7}{23}a^{5}-\frac{6}{23}a^{4}-\frac{4}{23}a^{2}+\frac{2}{23}a-\frac{10}{23}$, $\frac{1}{161}a^{15}+\frac{47}{161}a^{13}+\frac{1}{7}a^{12}+\frac{27}{161}a^{11}+\frac{36}{161}a^{10}-\frac{41}{161}a^{9}+\frac{11}{161}a^{8}+\frac{3}{23}a^{7}-\frac{1}{161}a^{6}-\frac{78}{161}a^{5}+\frac{4}{161}a^{4}-\frac{27}{161}a^{3}-\frac{26}{161}a^{2}-\frac{6}{23}a-\frac{1}{161}$, $\frac{1}{161}a^{16}-\frac{2}{161}a^{14}+\frac{44}{161}a^{13}-\frac{8}{161}a^{12}-\frac{41}{161}a^{11}-\frac{20}{161}a^{10}+\frac{32}{161}a^{9}-\frac{7}{23}a^{8}-\frac{50}{161}a^{7}+\frac{41}{161}a^{6}+\frac{25}{161}a^{5}-\frac{55}{161}a^{4}-\frac{26}{161}a^{3}-\frac{1}{23}a^{2}+\frac{62}{161}a+\frac{1}{23}$, $\frac{1}{161}a^{17}+\frac{2}{161}a^{14}+\frac{58}{161}a^{13}-\frac{2}{161}a^{12}-\frac{78}{161}a^{11}+\frac{76}{161}a^{10}+\frac{2}{161}a^{9}-\frac{6}{23}a^{8}+\frac{41}{161}a^{7}+\frac{79}{161}a^{6}-\frac{78}{161}a^{5}+\frac{73}{161}a^{4}-\frac{61}{161}a^{3}+\frac{17}{161}a^{2}-\frac{65}{161}$, $\frac{1}{3703}a^{18}+\frac{2}{3703}a^{17}+\frac{8}{3703}a^{15}-\frac{43}{3703}a^{14}-\frac{479}{3703}a^{13}+\frac{63}{529}a^{12}-\frac{1003}{3703}a^{11}-\frac{988}{3703}a^{10}-\frac{998}{3703}a^{9}+\frac{1276}{3703}a^{8}+\frac{233}{529}a^{7}+\frac{697}{3703}a^{6}+\frac{71}{161}a^{5}-\frac{1676}{3703}a^{4}+\frac{1343}{3703}a^{3}+\frac{1747}{3703}a^{2}+\frac{117}{3703}a-\frac{535}{3703}$, $\frac{1}{10\!\cdots\!81}a^{19}-\frac{13\!\cdots\!89}{10\!\cdots\!81}a^{18}-\frac{28\!\cdots\!54}{10\!\cdots\!81}a^{17}-\frac{89\!\cdots\!82}{10\!\cdots\!81}a^{16}+\frac{27\!\cdots\!53}{10\!\cdots\!81}a^{15}+\frac{47\!\cdots\!04}{10\!\cdots\!81}a^{14}-\frac{49\!\cdots\!20}{15\!\cdots\!83}a^{13}+\frac{46\!\cdots\!67}{10\!\cdots\!81}a^{12}+\frac{52\!\cdots\!45}{10\!\cdots\!81}a^{11}+\frac{16\!\cdots\!94}{47\!\cdots\!47}a^{10}-\frac{31\!\cdots\!85}{10\!\cdots\!81}a^{9}+\frac{10\!\cdots\!71}{10\!\cdots\!81}a^{8}-\frac{26\!\cdots\!30}{10\!\cdots\!81}a^{7}-\frac{36\!\cdots\!24}{10\!\cdots\!81}a^{6}+\frac{15\!\cdots\!32}{10\!\cdots\!81}a^{5}+\frac{22\!\cdots\!66}{10\!\cdots\!81}a^{4}-\frac{39\!\cdots\!90}{10\!\cdots\!81}a^{3}+\frac{43\!\cdots\!10}{10\!\cdots\!81}a^{2}+\frac{39\!\cdots\!88}{10\!\cdots\!81}a+\frac{54\!\cdots\!12}{10\!\cdots\!81}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{8221}$, which has order $8221$ (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: |
$\frac{80\!\cdots\!26}{15\!\cdots\!83}a^{19}+\frac{27\!\cdots\!64}{15\!\cdots\!83}a^{18}-\frac{14\!\cdots\!28}{15\!\cdots\!83}a^{17}-\frac{75\!\cdots\!30}{15\!\cdots\!83}a^{16}-\frac{16\!\cdots\!44}{15\!\cdots\!83}a^{15}-\frac{33\!\cdots\!92}{15\!\cdots\!83}a^{14}+\frac{17\!\cdots\!58}{15\!\cdots\!83}a^{13}+\frac{66\!\cdots\!80}{15\!\cdots\!83}a^{12}+\frac{66\!\cdots\!50}{15\!\cdots\!83}a^{11}+\frac{10\!\cdots\!60}{15\!\cdots\!83}a^{10}-\frac{21\!\cdots\!06}{15\!\cdots\!83}a^{9}+\frac{78\!\cdots\!69}{15\!\cdots\!83}a^{8}-\frac{14\!\cdots\!00}{15\!\cdots\!83}a^{7}+\frac{48\!\cdots\!74}{15\!\cdots\!83}a^{6}-\frac{60\!\cdots\!48}{15\!\cdots\!83}a^{5}+\frac{26\!\cdots\!83}{15\!\cdots\!83}a^{4}-\frac{40\!\cdots\!24}{15\!\cdots\!83}a^{3}+\frac{29\!\cdots\!09}{67\!\cdots\!21}a^{2}+\frac{30\!\cdots\!05}{15\!\cdots\!83}a+\frac{73\!\cdots\!36}{15\!\cdots\!83}$, $\frac{27\!\cdots\!80}{15\!\cdots\!83}a^{19}-\frac{12\!\cdots\!92}{15\!\cdots\!83}a^{18}+\frac{34\!\cdots\!76}{15\!\cdots\!83}a^{17}-\frac{25\!\cdots\!31}{15\!\cdots\!83}a^{16}+\frac{43\!\cdots\!44}{15\!\cdots\!83}a^{15}-\frac{20\!\cdots\!76}{15\!\cdots\!83}a^{14}+\frac{12\!\cdots\!04}{15\!\cdots\!83}a^{13}-\frac{28\!\cdots\!62}{15\!\cdots\!83}a^{12}+\frac{13\!\cdots\!12}{15\!\cdots\!83}a^{11}-\frac{45\!\cdots\!30}{15\!\cdots\!83}a^{10}+\frac{10\!\cdots\!14}{15\!\cdots\!83}a^{9}-\frac{36\!\cdots\!35}{15\!\cdots\!83}a^{8}+\frac{87\!\cdots\!84}{15\!\cdots\!83}a^{7}-\frac{17\!\cdots\!70}{15\!\cdots\!83}a^{6}+\frac{38\!\cdots\!06}{15\!\cdots\!83}a^{5}-\frac{50\!\cdots\!43}{15\!\cdots\!83}a^{4}+\frac{31\!\cdots\!42}{15\!\cdots\!83}a^{3}-\frac{16\!\cdots\!67}{15\!\cdots\!83}a^{2}+\frac{36\!\cdots\!28}{15\!\cdots\!83}a+\frac{19\!\cdots\!34}{15\!\cdots\!83}$, $\frac{39\!\cdots\!78}{15\!\cdots\!83}a^{19}-\frac{85\!\cdots\!77}{15\!\cdots\!83}a^{18}+\frac{51\!\cdots\!16}{15\!\cdots\!83}a^{17}-\frac{37\!\cdots\!64}{15\!\cdots\!83}a^{16}+\frac{12\!\cdots\!57}{15\!\cdots\!83}a^{15}-\frac{38\!\cdots\!88}{15\!\cdots\!83}a^{14}+\frac{22\!\cdots\!29}{15\!\cdots\!83}a^{13}-\frac{69\!\cdots\!58}{15\!\cdots\!83}a^{12}+\frac{25\!\cdots\!15}{15\!\cdots\!83}a^{11}-\frac{94\!\cdots\!46}{15\!\cdots\!83}a^{10}+\frac{25\!\cdots\!43}{15\!\cdots\!83}a^{9}-\frac{74\!\cdots\!30}{15\!\cdots\!83}a^{8}+\frac{20\!\cdots\!69}{15\!\cdots\!83}a^{7}-\frac{42\!\cdots\!83}{15\!\cdots\!83}a^{6}+\frac{88\!\cdots\!00}{15\!\cdots\!83}a^{5}-\frac{14\!\cdots\!67}{15\!\cdots\!83}a^{4}+\frac{12\!\cdots\!05}{15\!\cdots\!83}a^{3}-\frac{44\!\cdots\!55}{15\!\cdots\!83}a^{2}+\frac{68\!\cdots\!90}{15\!\cdots\!83}a-\frac{35\!\cdots\!69}{15\!\cdots\!83}$, $\frac{32\!\cdots\!40}{15\!\cdots\!83}a^{19}-\frac{39\!\cdots\!80}{15\!\cdots\!83}a^{18}+\frac{32\!\cdots\!89}{15\!\cdots\!83}a^{17}-\frac{30\!\cdots\!02}{15\!\cdots\!83}a^{16}+\frac{73\!\cdots\!85}{15\!\cdots\!83}a^{15}-\frac{26\!\cdots\!77}{15\!\cdots\!83}a^{14}+\frac{16\!\cdots\!07}{15\!\cdots\!83}a^{13}-\frac{18\!\cdots\!21}{67\!\cdots\!21}a^{12}+\frac{17\!\cdots\!87}{15\!\cdots\!83}a^{11}-\frac{63\!\cdots\!59}{15\!\cdots\!83}a^{10}+\frac{15\!\cdots\!00}{15\!\cdots\!83}a^{9}-\frac{50\!\cdots\!07}{15\!\cdots\!83}a^{8}+\frac{55\!\cdots\!79}{67\!\cdots\!21}a^{7}-\frac{25\!\cdots\!51}{15\!\cdots\!83}a^{6}+\frac{56\!\cdots\!38}{15\!\cdots\!83}a^{5}-\frac{82\!\cdots\!85}{15\!\cdots\!83}a^{4}+\frac{62\!\cdots\!11}{15\!\cdots\!83}a^{3}-\frac{25\!\cdots\!31}{15\!\cdots\!83}a^{2}+\frac{48\!\cdots\!81}{15\!\cdots\!83}a-\frac{82\!\cdots\!15}{15\!\cdots\!83}$, $\frac{22\!\cdots\!66}{48\!\cdots\!13}a^{19}-\frac{48\!\cdots\!33}{33\!\cdots\!91}a^{18}-\frac{46\!\cdots\!23}{33\!\cdots\!91}a^{17}-\frac{11\!\cdots\!43}{33\!\cdots\!91}a^{16}+\frac{63\!\cdots\!44}{33\!\cdots\!91}a^{15}-\frac{11\!\cdots\!57}{33\!\cdots\!91}a^{14}+\frac{61\!\cdots\!88}{33\!\cdots\!91}a^{13}-\frac{26\!\cdots\!31}{33\!\cdots\!91}a^{12}+\frac{75\!\cdots\!56}{33\!\cdots\!91}a^{11}-\frac{26\!\cdots\!45}{33\!\cdots\!91}a^{10}+\frac{76\!\cdots\!15}{33\!\cdots\!91}a^{9}-\frac{17\!\cdots\!77}{33\!\cdots\!91}a^{8}+\frac{44\!\cdots\!82}{33\!\cdots\!91}a^{7}-\frac{92\!\cdots\!02}{33\!\cdots\!91}a^{6}+\frac{10\!\cdots\!66}{33\!\cdots\!91}a^{5}-\frac{29\!\cdots\!51}{14\!\cdots\!17}a^{4}+\frac{34\!\cdots\!48}{33\!\cdots\!91}a^{3}-\frac{35\!\cdots\!26}{48\!\cdots\!13}a^{2}+\frac{11\!\cdots\!93}{33\!\cdots\!91}a-\frac{65\!\cdots\!88}{33\!\cdots\!91}$, $\frac{11\!\cdots\!68}{10\!\cdots\!81}a^{19}+\frac{41\!\cdots\!86}{10\!\cdots\!81}a^{18}+\frac{33\!\cdots\!88}{10\!\cdots\!81}a^{17}-\frac{10\!\cdots\!92}{10\!\cdots\!81}a^{16}+\frac{90\!\cdots\!62}{10\!\cdots\!81}a^{15}-\frac{93\!\cdots\!09}{15\!\cdots\!83}a^{14}+\frac{45\!\cdots\!73}{10\!\cdots\!81}a^{13}-\frac{69\!\cdots\!85}{10\!\cdots\!81}a^{12}+\frac{44\!\cdots\!15}{10\!\cdots\!81}a^{11}-\frac{19\!\cdots\!27}{15\!\cdots\!83}a^{10}+\frac{38\!\cdots\!70}{15\!\cdots\!83}a^{9}-\frac{11\!\cdots\!02}{10\!\cdots\!81}a^{8}+\frac{22\!\cdots\!28}{10\!\cdots\!81}a^{7}-\frac{38\!\cdots\!84}{10\!\cdots\!81}a^{6}+\frac{14\!\cdots\!10}{15\!\cdots\!83}a^{5}-\frac{68\!\cdots\!08}{10\!\cdots\!81}a^{4}-\frac{38\!\cdots\!52}{10\!\cdots\!81}a^{3}-\frac{28\!\cdots\!49}{10\!\cdots\!81}a^{2}+\frac{12\!\cdots\!58}{10\!\cdots\!81}a-\frac{22\!\cdots\!61}{10\!\cdots\!81}$, $\frac{25\!\cdots\!84}{10\!\cdots\!81}a^{19}+\frac{36\!\cdots\!24}{10\!\cdots\!81}a^{18}+\frac{12\!\cdots\!23}{10\!\cdots\!81}a^{17}-\frac{24\!\cdots\!22}{10\!\cdots\!81}a^{16}+\frac{26\!\cdots\!71}{10\!\cdots\!81}a^{15}-\frac{16\!\cdots\!49}{10\!\cdots\!81}a^{14}+\frac{11\!\cdots\!35}{10\!\cdots\!81}a^{13}-\frac{18\!\cdots\!62}{10\!\cdots\!81}a^{12}+\frac{15\!\cdots\!01}{15\!\cdots\!83}a^{11}-\frac{50\!\cdots\!29}{15\!\cdots\!83}a^{10}+\frac{72\!\cdots\!65}{10\!\cdots\!81}a^{9}-\frac{28\!\cdots\!81}{10\!\cdots\!81}a^{8}+\frac{60\!\cdots\!49}{10\!\cdots\!81}a^{7}-\frac{15\!\cdots\!20}{15\!\cdots\!83}a^{6}+\frac{26\!\cdots\!50}{10\!\cdots\!81}a^{5}-\frac{34\!\cdots\!14}{15\!\cdots\!83}a^{4}+\frac{68\!\cdots\!04}{10\!\cdots\!81}a^{3}-\frac{87\!\cdots\!37}{10\!\cdots\!81}a^{2}+\frac{30\!\cdots\!16}{10\!\cdots\!81}a-\frac{13\!\cdots\!09}{10\!\cdots\!81}$, $\frac{89\!\cdots\!54}{47\!\cdots\!47}a^{19}-\frac{17\!\cdots\!80}{10\!\cdots\!81}a^{18}+\frac{12\!\cdots\!77}{15\!\cdots\!83}a^{17}-\frac{84\!\cdots\!06}{47\!\cdots\!47}a^{16}+\frac{18\!\cdots\!92}{10\!\cdots\!81}a^{15}-\frac{40\!\cdots\!06}{10\!\cdots\!81}a^{14}+\frac{29\!\cdots\!14}{15\!\cdots\!83}a^{13}-\frac{13\!\cdots\!87}{15\!\cdots\!83}a^{12}+\frac{25\!\cdots\!37}{10\!\cdots\!81}a^{11}-\frac{15\!\cdots\!65}{15\!\cdots\!83}a^{10}+\frac{33\!\cdots\!12}{10\!\cdots\!81}a^{9}-\frac{84\!\cdots\!29}{10\!\cdots\!81}a^{8}+\frac{37\!\cdots\!81}{15\!\cdots\!83}a^{7}-\frac{84\!\cdots\!79}{15\!\cdots\!83}a^{6}+\frac{49\!\cdots\!00}{47\!\cdots\!47}a^{5}-\frac{33\!\cdots\!25}{15\!\cdots\!83}a^{4}+\frac{25\!\cdots\!81}{10\!\cdots\!81}a^{3}-\frac{66\!\cdots\!32}{10\!\cdots\!81}a^{2}+\frac{67\!\cdots\!92}{10\!\cdots\!81}a-\frac{76\!\cdots\!74}{10\!\cdots\!81}$, $\frac{23\!\cdots\!28}{10\!\cdots\!81}a^{19}+\frac{30\!\cdots\!72}{10\!\cdots\!81}a^{18}+\frac{87\!\cdots\!50}{10\!\cdots\!81}a^{17}-\frac{20\!\cdots\!30}{10\!\cdots\!81}a^{16}+\frac{27\!\cdots\!41}{10\!\cdots\!81}a^{15}-\frac{17\!\cdots\!05}{10\!\cdots\!81}a^{14}+\frac{76\!\cdots\!41}{10\!\cdots\!81}a^{13}-\frac{16\!\cdots\!00}{15\!\cdots\!83}a^{12}+\frac{97\!\cdots\!34}{10\!\cdots\!81}a^{11}-\frac{21\!\cdots\!98}{10\!\cdots\!81}a^{10}+\frac{58\!\cdots\!50}{10\!\cdots\!81}a^{9}-\frac{21\!\cdots\!35}{10\!\cdots\!81}a^{8}+\frac{37\!\cdots\!58}{10\!\cdots\!81}a^{7}-\frac{87\!\cdots\!40}{10\!\cdots\!81}a^{6}+\frac{17\!\cdots\!38}{10\!\cdots\!81}a^{5}-\frac{14\!\cdots\!71}{10\!\cdots\!81}a^{4}+\frac{16\!\cdots\!34}{47\!\cdots\!47}a^{3}-\frac{50\!\cdots\!00}{10\!\cdots\!81}a^{2}+\frac{19\!\cdots\!10}{10\!\cdots\!81}a-\frac{24\!\cdots\!81}{10\!\cdots\!81}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 15197121.6751 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 15197121.6751 \cdot 8221}{2\cdot\sqrt{396508891893913150393870750167904820789}}\cr\approx \mathstrut & 0.300834895640 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113)
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 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113, 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 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113);
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 + x^18 - 93*x^17 + 209*x^16 - 800*x^15 + 4981*x^14 - 12556*x^13 + 52672*x^12 - 186410*x^11 + 461402*x^10 - 1486634*x^9 + 3730667*x^8 - 7453323*x^7 + 16223240*x^6 - 23433182*x^5 + 16658019*x^4 - 7158636*x^3 + 5906450*x^2 - 2576267*x + 1575113);
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
| \(\Q(\sqrt{29}) \), 4.0.24389.1, \(\Q(\zeta_{11})^+\), 10.10.4396746947664269.1 |
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 | $20$ | $20$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/23.1.0.1}{1} }^{20}$ | R | $20$ | $20$ | $20$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 | $[\ ]_{5}^{4}$ |
|
\(29\)
| 29.20.15.2 | $x^{20} + 157 x^{16} + 108 x^{15} - 1976 x^{12} - 202608 x^{11} + 4374 x^{10} + 277928 x^{8} + 14926896 x^{7} + 7214184 x^{6} + 78732 x^{5} + 13300496 x^{4} - 99140544 x^{3} + 66210696 x^{2} - 11179944 x + 13783745$ | $4$ | $5$ | $15$ | 20T1 | $[\ ]_{4}^{5}$ |