Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 7*y^18 + 7*y^17 - 42*y^16 + 73*y^15 - 30*y^14 - 116*y^13 + 243*y^12 - 217*y^11 - 7*y^10 + 283*y^9 - 285*y^8 + 50*y^7 + 212*y^6 - 381*y^5 + 244*y^4 - 117*y^3 + 73*y^2 - 17*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*x + 1)
\( x^{20} - 5 x^{19} + 7 x^{18} + 7 x^{17} - 42 x^{16} + 73 x^{15} - 30 x^{14} - 116 x^{13} + 243 x^{12} + \cdots + 1 \)
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: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6802209109663753569286129\)
\(\medspace = 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.44\) | 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: | $11^{4/5}23^{1/2}\approx 32.65713384043754$ | ||
| Ramified primes: |
\(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{33\!\cdots\!93}a^{19}-\frac{13\!\cdots\!34}{33\!\cdots\!93}a^{18}-\frac{10\!\cdots\!00}{33\!\cdots\!93}a^{17}+\frac{12\!\cdots\!76}{33\!\cdots\!93}a^{16}+\frac{85\!\cdots\!04}{33\!\cdots\!93}a^{15}+\frac{14\!\cdots\!30}{33\!\cdots\!93}a^{14}+\frac{73\!\cdots\!58}{33\!\cdots\!93}a^{13}-\frac{73\!\cdots\!13}{33\!\cdots\!93}a^{12}+\frac{39\!\cdots\!29}{33\!\cdots\!93}a^{11}+\frac{74\!\cdots\!89}{33\!\cdots\!93}a^{10}-\frac{10\!\cdots\!22}{33\!\cdots\!93}a^{9}+\frac{25\!\cdots\!12}{33\!\cdots\!93}a^{8}+\frac{15\!\cdots\!95}{33\!\cdots\!93}a^{7}+\frac{79\!\cdots\!52}{33\!\cdots\!93}a^{6}+\frac{15\!\cdots\!87}{33\!\cdots\!93}a^{5}+\frac{86\!\cdots\!66}{33\!\cdots\!93}a^{4}+\frac{12\!\cdots\!11}{33\!\cdots\!93}a^{3}+\frac{10\!\cdots\!70}{33\!\cdots\!93}a^{2}-\frac{43\!\cdots\!85}{33\!\cdots\!93}a+\frac{56\!\cdots\!86}{33\!\cdots\!93}$
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 $1$
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: | $11$ | 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: |
$\frac{82\!\cdots\!22}{33\!\cdots\!93}a^{19}-\frac{38\!\cdots\!10}{33\!\cdots\!93}a^{18}+\frac{45\!\cdots\!94}{33\!\cdots\!93}a^{17}+\frac{74\!\cdots\!12}{33\!\cdots\!93}a^{16}-\frac{32\!\cdots\!72}{33\!\cdots\!93}a^{15}+\frac{49\!\cdots\!32}{33\!\cdots\!93}a^{14}-\frac{74\!\cdots\!38}{33\!\cdots\!93}a^{13}-\frac{10\!\cdots\!45}{33\!\cdots\!93}a^{12}+\frac{16\!\cdots\!37}{33\!\cdots\!93}a^{11}-\frac{12\!\cdots\!30}{33\!\cdots\!93}a^{10}-\frac{53\!\cdots\!39}{33\!\cdots\!93}a^{9}+\frac{22\!\cdots\!15}{33\!\cdots\!93}a^{8}-\frac{16\!\cdots\!37}{33\!\cdots\!93}a^{7}-\frac{25\!\cdots\!97}{33\!\cdots\!93}a^{6}+\frac{17\!\cdots\!88}{33\!\cdots\!93}a^{5}-\frac{25\!\cdots\!42}{33\!\cdots\!93}a^{4}+\frac{10\!\cdots\!43}{33\!\cdots\!93}a^{3}-\frac{43\!\cdots\!30}{33\!\cdots\!93}a^{2}+\frac{40\!\cdots\!99}{33\!\cdots\!93}a-\frac{15\!\cdots\!86}{33\!\cdots\!93}$, $\frac{11\!\cdots\!64}{33\!\cdots\!93}a^{19}-\frac{54\!\cdots\!73}{33\!\cdots\!93}a^{18}+\frac{69\!\cdots\!88}{33\!\cdots\!93}a^{17}+\frac{89\!\cdots\!94}{33\!\cdots\!93}a^{16}-\frac{45\!\cdots\!03}{33\!\cdots\!93}a^{15}+\frac{74\!\cdots\!43}{33\!\cdots\!93}a^{14}-\frac{21\!\cdots\!96}{33\!\cdots\!93}a^{13}-\frac{13\!\cdots\!51}{33\!\cdots\!93}a^{12}+\frac{25\!\cdots\!00}{33\!\cdots\!93}a^{11}-\frac{20\!\cdots\!90}{33\!\cdots\!93}a^{10}-\frac{37\!\cdots\!41}{33\!\cdots\!93}a^{9}+\frac{30\!\cdots\!31}{33\!\cdots\!93}a^{8}-\frac{26\!\cdots\!57}{33\!\cdots\!93}a^{7}+\frac{18\!\cdots\!64}{33\!\cdots\!93}a^{6}+\frac{23\!\cdots\!49}{33\!\cdots\!93}a^{5}-\frac{39\!\cdots\!89}{33\!\cdots\!93}a^{4}+\frac{21\!\cdots\!67}{33\!\cdots\!93}a^{3}-\frac{10\!\cdots\!34}{33\!\cdots\!93}a^{2}+\frac{66\!\cdots\!47}{33\!\cdots\!93}a-\frac{77\!\cdots\!64}{33\!\cdots\!93}$, $\frac{11\!\cdots\!10}{33\!\cdots\!93}a^{19}-\frac{55\!\cdots\!15}{33\!\cdots\!93}a^{18}+\frac{72\!\cdots\!57}{33\!\cdots\!93}a^{17}+\frac{85\!\cdots\!50}{33\!\cdots\!93}a^{16}-\frac{45\!\cdots\!41}{33\!\cdots\!93}a^{15}+\frac{77\!\cdots\!29}{33\!\cdots\!93}a^{14}-\frac{26\!\cdots\!87}{33\!\cdots\!93}a^{13}-\frac{13\!\cdots\!40}{33\!\cdots\!93}a^{12}+\frac{25\!\cdots\!47}{33\!\cdots\!93}a^{11}-\frac{22\!\cdots\!62}{33\!\cdots\!93}a^{10}-\frac{21\!\cdots\!07}{33\!\cdots\!93}a^{9}+\frac{30\!\cdots\!90}{33\!\cdots\!93}a^{8}-\frac{29\!\cdots\!87}{33\!\cdots\!93}a^{7}+\frac{43\!\cdots\!80}{33\!\cdots\!93}a^{6}+\frac{23\!\cdots\!41}{33\!\cdots\!93}a^{5}-\frac{40\!\cdots\!98}{33\!\cdots\!93}a^{4}+\frac{24\!\cdots\!61}{33\!\cdots\!93}a^{3}-\frac{12\!\cdots\!38}{33\!\cdots\!93}a^{2}+\frac{70\!\cdots\!85}{33\!\cdots\!93}a-\frac{90\!\cdots\!40}{33\!\cdots\!93}$, $\frac{83\!\cdots\!88}{33\!\cdots\!93}a^{19}-\frac{40\!\cdots\!99}{33\!\cdots\!93}a^{18}+\frac{52\!\cdots\!96}{33\!\cdots\!93}a^{17}+\frac{63\!\cdots\!89}{33\!\cdots\!93}a^{16}-\frac{33\!\cdots\!99}{33\!\cdots\!93}a^{15}+\frac{56\!\cdots\!15}{33\!\cdots\!93}a^{14}-\frac{18\!\cdots\!42}{33\!\cdots\!93}a^{13}-\frac{95\!\cdots\!95}{33\!\cdots\!93}a^{12}+\frac{18\!\cdots\!54}{33\!\cdots\!93}a^{11}-\frac{16\!\cdots\!51}{33\!\cdots\!93}a^{10}-\frac{17\!\cdots\!42}{33\!\cdots\!93}a^{9}+\frac{22\!\cdots\!04}{33\!\cdots\!93}a^{8}-\frac{21\!\cdots\!01}{33\!\cdots\!93}a^{7}+\frac{28\!\cdots\!56}{33\!\cdots\!93}a^{6}+\frac{17\!\cdots\!96}{33\!\cdots\!93}a^{5}-\frac{29\!\cdots\!60}{33\!\cdots\!93}a^{4}+\frac{17\!\cdots\!75}{33\!\cdots\!93}a^{3}-\frac{89\!\cdots\!58}{33\!\cdots\!93}a^{2}+\frac{50\!\cdots\!06}{33\!\cdots\!93}a-\frac{67\!\cdots\!17}{33\!\cdots\!93}$, $\frac{10\!\cdots\!48}{33\!\cdots\!93}a^{19}-\frac{50\!\cdots\!12}{33\!\cdots\!93}a^{18}+\frac{62\!\cdots\!81}{33\!\cdots\!93}a^{17}+\frac{88\!\cdots\!15}{33\!\cdots\!93}a^{16}-\frac{42\!\cdots\!59}{33\!\cdots\!93}a^{15}+\frac{68\!\cdots\!83}{33\!\cdots\!93}a^{14}-\frac{16\!\cdots\!13}{33\!\cdots\!93}a^{13}-\frac{12\!\cdots\!40}{33\!\cdots\!93}a^{12}+\frac{23\!\cdots\!21}{33\!\cdots\!93}a^{11}-\frac{18\!\cdots\!90}{33\!\cdots\!93}a^{10}-\frac{47\!\cdots\!75}{33\!\cdots\!93}a^{9}+\frac{29\!\cdots\!84}{33\!\cdots\!93}a^{8}-\frac{23\!\cdots\!95}{33\!\cdots\!93}a^{7}-\frac{15\!\cdots\!46}{33\!\cdots\!93}a^{6}+\frac{22\!\cdots\!43}{33\!\cdots\!93}a^{5}-\frac{35\!\cdots\!48}{33\!\cdots\!93}a^{4}+\frac{17\!\cdots\!79}{33\!\cdots\!93}a^{3}-\frac{82\!\cdots\!74}{33\!\cdots\!93}a^{2}+\frac{58\!\cdots\!28}{33\!\cdots\!93}a-\frac{84\!\cdots\!58}{33\!\cdots\!93}$, $\frac{62\!\cdots\!38}{33\!\cdots\!93}a^{19}-\frac{31\!\cdots\!39}{33\!\cdots\!93}a^{18}+\frac{44\!\cdots\!29}{33\!\cdots\!93}a^{17}+\frac{42\!\cdots\!52}{33\!\cdots\!93}a^{16}-\frac{26\!\cdots\!01}{33\!\cdots\!93}a^{15}+\frac{46\!\cdots\!10}{33\!\cdots\!93}a^{14}-\frac{19\!\cdots\!80}{33\!\cdots\!93}a^{13}-\frac{71\!\cdots\!89}{33\!\cdots\!93}a^{12}+\frac{15\!\cdots\!45}{33\!\cdots\!93}a^{11}-\frac{13\!\cdots\!78}{33\!\cdots\!93}a^{10}-\frac{48\!\cdots\!23}{33\!\cdots\!93}a^{9}+\frac{17\!\cdots\!36}{33\!\cdots\!93}a^{8}-\frac{18\!\cdots\!93}{33\!\cdots\!93}a^{7}+\frac{36\!\cdots\!35}{33\!\cdots\!93}a^{6}+\frac{12\!\cdots\!29}{33\!\cdots\!93}a^{5}-\frac{23\!\cdots\!33}{33\!\cdots\!93}a^{4}+\frac{16\!\cdots\!30}{33\!\cdots\!93}a^{3}-\frac{79\!\cdots\!04}{33\!\cdots\!93}a^{2}+\frac{50\!\cdots\!24}{33\!\cdots\!93}a-\frac{10\!\cdots\!38}{33\!\cdots\!93}$, $\frac{57\!\cdots\!01}{33\!\cdots\!93}a^{19}-\frac{27\!\cdots\!01}{33\!\cdots\!93}a^{18}+\frac{34\!\cdots\!87}{33\!\cdots\!93}a^{17}+\frac{46\!\cdots\!12}{33\!\cdots\!93}a^{16}-\frac{22\!\cdots\!58}{33\!\cdots\!93}a^{15}+\frac{37\!\cdots\!69}{33\!\cdots\!93}a^{14}-\frac{10\!\cdots\!60}{33\!\cdots\!93}a^{13}-\frac{67\!\cdots\!10}{33\!\cdots\!93}a^{12}+\frac{12\!\cdots\!97}{33\!\cdots\!93}a^{11}-\frac{10\!\cdots\!15}{33\!\cdots\!93}a^{10}-\frac{21\!\cdots\!09}{33\!\cdots\!93}a^{9}+\frac{15\!\cdots\!92}{33\!\cdots\!93}a^{8}-\frac{13\!\cdots\!67}{33\!\cdots\!93}a^{7}+\frac{51\!\cdots\!24}{33\!\cdots\!93}a^{6}+\frac{12\!\cdots\!70}{33\!\cdots\!93}a^{5}-\frac{19\!\cdots\!64}{33\!\cdots\!93}a^{4}+\frac{99\!\cdots\!78}{33\!\cdots\!93}a^{3}-\frac{46\!\cdots\!37}{33\!\cdots\!93}a^{2}+\frac{32\!\cdots\!91}{33\!\cdots\!93}a-\frac{46\!\cdots\!38}{33\!\cdots\!93}$, $\frac{26\!\cdots\!00}{33\!\cdots\!93}a^{19}-\frac{10\!\cdots\!32}{33\!\cdots\!93}a^{18}+\frac{71\!\cdots\!72}{33\!\cdots\!93}a^{17}+\frac{32\!\cdots\!39}{33\!\cdots\!93}a^{16}-\frac{90\!\cdots\!15}{33\!\cdots\!93}a^{15}+\frac{97\!\cdots\!82}{33\!\cdots\!93}a^{14}+\frac{73\!\cdots\!59}{33\!\cdots\!93}a^{13}-\frac{34\!\cdots\!67}{33\!\cdots\!93}a^{12}+\frac{36\!\cdots\!71}{33\!\cdots\!93}a^{11}-\frac{58\!\cdots\!52}{33\!\cdots\!93}a^{10}-\frac{43\!\cdots\!82}{33\!\cdots\!93}a^{9}+\frac{67\!\cdots\!26}{33\!\cdots\!93}a^{8}-\frac{11\!\cdots\!06}{33\!\cdots\!93}a^{7}-\frac{42\!\cdots\!22}{33\!\cdots\!93}a^{6}+\frac{60\!\cdots\!65}{33\!\cdots\!93}a^{5}-\frac{53\!\cdots\!50}{33\!\cdots\!93}a^{4}-\frac{18\!\cdots\!99}{33\!\cdots\!93}a^{3}+\frac{14\!\cdots\!14}{33\!\cdots\!93}a^{2}-\frac{21\!\cdots\!06}{33\!\cdots\!93}a+\frac{97\!\cdots\!46}{33\!\cdots\!93}$, $\frac{11\!\cdots\!98}{33\!\cdots\!93}a^{19}-\frac{57\!\cdots\!97}{33\!\cdots\!93}a^{18}+\frac{73\!\cdots\!87}{33\!\cdots\!93}a^{17}+\frac{95\!\cdots\!87}{33\!\cdots\!93}a^{16}-\frac{48\!\cdots\!48}{33\!\cdots\!93}a^{15}+\frac{78\!\cdots\!94}{33\!\cdots\!93}a^{14}-\frac{21\!\cdots\!27}{33\!\cdots\!93}a^{13}-\frac{14\!\cdots\!94}{33\!\cdots\!93}a^{12}+\frac{26\!\cdots\!57}{33\!\cdots\!93}a^{11}-\frac{20\!\cdots\!33}{33\!\cdots\!93}a^{10}-\frac{52\!\cdots\!49}{33\!\cdots\!93}a^{9}+\frac{33\!\cdots\!42}{33\!\cdots\!93}a^{8}-\frac{27\!\cdots\!73}{33\!\cdots\!93}a^{7}-\frac{20\!\cdots\!11}{33\!\cdots\!93}a^{6}+\frac{25\!\cdots\!05}{33\!\cdots\!93}a^{5}-\frac{40\!\cdots\!63}{33\!\cdots\!93}a^{4}+\frac{21\!\cdots\!84}{33\!\cdots\!93}a^{3}-\frac{88\!\cdots\!93}{33\!\cdots\!93}a^{2}+\frac{68\!\cdots\!47}{33\!\cdots\!93}a-\frac{60\!\cdots\!93}{33\!\cdots\!93}$, $\frac{80\!\cdots\!03}{33\!\cdots\!93}a^{19}-\frac{37\!\cdots\!29}{33\!\cdots\!93}a^{18}+\frac{44\!\cdots\!61}{33\!\cdots\!93}a^{17}+\frac{68\!\cdots\!45}{33\!\cdots\!93}a^{16}-\frac{31\!\cdots\!11}{33\!\cdots\!93}a^{15}+\frac{48\!\cdots\!84}{33\!\cdots\!93}a^{14}-\frac{98\!\cdots\!86}{33\!\cdots\!93}a^{13}-\frac{93\!\cdots\!97}{33\!\cdots\!93}a^{12}+\frac{16\!\cdots\!52}{33\!\cdots\!93}a^{11}-\frac{12\!\cdots\!04}{33\!\cdots\!93}a^{10}-\frac{37\!\cdots\!44}{33\!\cdots\!93}a^{9}+\frac{20\!\cdots\!32}{33\!\cdots\!93}a^{8}-\frac{16\!\cdots\!85}{33\!\cdots\!93}a^{7}+\frac{18\!\cdots\!32}{33\!\cdots\!93}a^{6}+\frac{16\!\cdots\!47}{33\!\cdots\!93}a^{5}-\frac{25\!\cdots\!03}{33\!\cdots\!93}a^{4}+\frac{12\!\cdots\!27}{33\!\cdots\!93}a^{3}-\frac{67\!\cdots\!04}{33\!\cdots\!93}a^{2}+\frac{34\!\cdots\!07}{33\!\cdots\!93}a-\frac{31\!\cdots\!26}{33\!\cdots\!93}$, $\frac{94\!\cdots\!69}{33\!\cdots\!93}a^{19}-\frac{45\!\cdots\!05}{33\!\cdots\!93}a^{18}+\frac{58\!\cdots\!64}{33\!\cdots\!93}a^{17}+\frac{76\!\cdots\!20}{33\!\cdots\!93}a^{16}-\frac{38\!\cdots\!72}{33\!\cdots\!93}a^{15}+\frac{62\!\cdots\!25}{33\!\cdots\!93}a^{14}-\frac{17\!\cdots\!46}{33\!\cdots\!93}a^{13}-\frac{11\!\cdots\!69}{33\!\cdots\!93}a^{12}+\frac{21\!\cdots\!87}{33\!\cdots\!93}a^{11}-\frac{17\!\cdots\!66}{33\!\cdots\!93}a^{10}-\frac{36\!\cdots\!11}{33\!\cdots\!93}a^{9}+\frac{26\!\cdots\!59}{33\!\cdots\!93}a^{8}-\frac{22\!\cdots\!09}{33\!\cdots\!93}a^{7}+\frac{79\!\cdots\!97}{33\!\cdots\!93}a^{6}+\frac{20\!\cdots\!75}{33\!\cdots\!93}a^{5}-\frac{32\!\cdots\!00}{33\!\cdots\!93}a^{4}+\frac{17\!\cdots\!99}{33\!\cdots\!93}a^{3}-\frac{78\!\cdots\!33}{33\!\cdots\!93}a^{2}+\frac{51\!\cdots\!75}{33\!\cdots\!93}a-\frac{80\!\cdots\!95}{33\!\cdots\!93}$
| 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: | \( 26285.5537069 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 26285.5537069 \cdot 1}{2\cdot\sqrt{6802209109663753569286129}}\cr\approx \mathstrut & 0.195848871945 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*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 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*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 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*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 - 5*x^19 + 7*x^18 + 7*x^17 - 42*x^16 + 73*x^15 - 30*x^14 - 116*x^13 + 243*x^12 - 217*x^11 - 7*x^10 + 283*x^9 - 285*x^8 + 50*x^7 + 212*x^6 - 381*x^5 + 244*x^4 - 117*x^3 + 73*x^2 - 17*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
$C_2\wr C_5$ (as 20T40):
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 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.4930254263.1, 10.8.2608104505127.1, 10.2.113395848049.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
| Minimal sibling: | 10.4.4930254263.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |