Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64)
gp: K = bnfinit(y^18 - 6*y^17 + 30*y^16 - 117*y^15 + 309*y^14 - 627*y^13 + 930*y^12 - 627*y^11 - 510*y^10 - 974*y^9 + 6243*y^8 - 2055*y^7 - 11115*y^6 + 10734*y^5 - 696*y^4 + 168*y^3 - 1344*y^2 - 192*y + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64)
\( x^{18} - 6 x^{17} + 30 x^{16} - 117 x^{15} + 309 x^{14} - 627 x^{13} + 930 x^{12} - 627 x^{11} + \cdots + 64 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(11360989554893559098699461641\)
\(\medspace = 3^{32}\cdot 19^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.19\) | 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: | $3^{16/9}19^{5/6}\approx 82.00561024067765$ | ||
| Ramified primes: |
\(3\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{12}-\frac{1}{2}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{12}a^{13}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{12}a^{4}+\frac{1}{3}a^{3}-\frac{5}{12}a^{2}+\frac{1}{3}$, $\frac{1}{36}a^{14}+\frac{1}{36}a^{13}+\frac{1}{36}a^{12}+\frac{1}{18}a^{11}-\frac{1}{36}a^{10}+\frac{1}{18}a^{9}+\frac{1}{6}a^{8}-\frac{1}{12}a^{7}-\frac{1}{12}a^{6}+\frac{17}{36}a^{5}-\frac{5}{18}a^{4}-\frac{1}{36}a^{3}+\frac{5}{18}a^{2}-\frac{7}{18}a-\frac{2}{9}$, $\frac{1}{72}a^{15}+\frac{1}{72}a^{12}-\frac{1}{24}a^{11}+\frac{1}{24}a^{10}-\frac{1}{36}a^{9}+\frac{1}{8}a^{8}+\frac{5}{18}a^{6}+\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{11}{72}a^{3}-\frac{1}{12}a^{2}-\frac{1}{6}a+\frac{4}{9}$, $\frac{1}{31392}a^{16}+\frac{65}{15696}a^{15}+\frac{181}{15696}a^{14}-\frac{93}{3488}a^{13}-\frac{289}{10464}a^{12}+\frac{2401}{31392}a^{11}-\frac{845}{15696}a^{10}+\frac{17}{31392}a^{9}-\frac{325}{5232}a^{8}-\frac{1505}{15696}a^{7}+\frac{11315}{31392}a^{6}+\frac{13777}{31392}a^{5}+\frac{4867}{10464}a^{4}-\frac{1589}{5232}a^{3}-\frac{61}{7848}a^{2}+\frac{559}{1962}a-\frac{455}{1962}$, $\frac{1}{10\!\cdots\!76}a^{17}-\frac{967386342095642}{94\!\cdots\!63}a^{16}-\frac{14\!\cdots\!77}{54\!\cdots\!88}a^{15}-\frac{30\!\cdots\!29}{10\!\cdots\!76}a^{14}+\frac{11\!\cdots\!15}{10\!\cdots\!76}a^{13}-\frac{80\!\cdots\!25}{10\!\cdots\!76}a^{12}+\frac{24\!\cdots\!27}{30\!\cdots\!16}a^{11}-\frac{79\!\cdots\!19}{10\!\cdots\!76}a^{10}+\frac{26\!\cdots\!55}{37\!\cdots\!52}a^{9}-\frac{10\!\cdots\!51}{54\!\cdots\!88}a^{8}+\frac{12\!\cdots\!61}{36\!\cdots\!92}a^{7}-\frac{37\!\cdots\!33}{10\!\cdots\!76}a^{6}+\frac{53\!\cdots\!19}{10\!\cdots\!76}a^{5}-\frac{32\!\cdots\!33}{13\!\cdots\!72}a^{4}+\frac{42\!\cdots\!19}{13\!\cdots\!72}a^{3}+\frac{15\!\cdots\!57}{45\!\cdots\!24}a^{2}-\frac{24\!\cdots\!51}{67\!\cdots\!36}a-\frac{44\!\cdots\!17}{11\!\cdots\!56}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: | $11$ | 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: |
$\frac{77\!\cdots\!41}{99\!\cdots\!64}a^{17}-\frac{58\!\cdots\!69}{82\!\cdots\!72}a^{16}+\frac{47\!\cdots\!91}{49\!\cdots\!32}a^{15}-\frac{46\!\cdots\!81}{99\!\cdots\!64}a^{14}+\frac{67\!\cdots\!53}{33\!\cdots\!88}a^{13}-\frac{68\!\cdots\!81}{99\!\cdots\!64}a^{12}+\frac{24\!\cdots\!51}{15\!\cdots\!26}a^{11}-\frac{31\!\cdots\!55}{11\!\cdots\!96}a^{10}+\frac{84\!\cdots\!65}{24\!\cdots\!16}a^{9}-\frac{32\!\cdots\!39}{49\!\cdots\!32}a^{8}-\frac{11\!\cdots\!19}{33\!\cdots\!88}a^{7}-\frac{99\!\cdots\!29}{99\!\cdots\!64}a^{6}+\frac{29\!\cdots\!51}{99\!\cdots\!64}a^{5}+\frac{11\!\cdots\!09}{82\!\cdots\!72}a^{4}-\frac{88\!\cdots\!55}{15\!\cdots\!26}a^{3}+\frac{27\!\cdots\!89}{12\!\cdots\!08}a^{2}+\frac{36\!\cdots\!29}{69\!\cdots\!56}a+\frac{16\!\cdots\!03}{31\!\cdots\!52}$, $\frac{18\!\cdots\!55}{49\!\cdots\!32}a^{17}-\frac{52\!\cdots\!07}{82\!\cdots\!72}a^{16}+\frac{35\!\cdots\!75}{24\!\cdots\!16}a^{15}-\frac{36\!\cdots\!47}{49\!\cdots\!32}a^{14}+\frac{56\!\cdots\!05}{16\!\cdots\!44}a^{13}-\frac{60\!\cdots\!25}{49\!\cdots\!32}a^{12}+\frac{71\!\cdots\!21}{24\!\cdots\!16}a^{11}-\frac{88\!\cdots\!51}{16\!\cdots\!44}a^{10}+\frac{16\!\cdots\!27}{24\!\cdots\!16}a^{9}-\frac{50\!\cdots\!59}{24\!\cdots\!16}a^{8}-\frac{10\!\cdots\!89}{16\!\cdots\!44}a^{7}-\frac{89\!\cdots\!37}{49\!\cdots\!32}a^{6}+\frac{27\!\cdots\!35}{49\!\cdots\!32}a^{5}+\frac{21\!\cdots\!61}{82\!\cdots\!72}a^{4}-\frac{13\!\cdots\!71}{12\!\cdots\!08}a^{3}+\frac{13\!\cdots\!23}{31\!\cdots\!52}a^{2}+\frac{10\!\cdots\!21}{10\!\cdots\!84}a+\frac{12\!\cdots\!74}{77\!\cdots\!63}$, $\frac{41\!\cdots\!77}{10\!\cdots\!76}a^{17}-\frac{77\!\cdots\!49}{16\!\cdots\!34}a^{16}+\frac{12\!\cdots\!63}{54\!\cdots\!88}a^{15}-\frac{37\!\cdots\!63}{36\!\cdots\!92}a^{14}+\frac{36\!\cdots\!03}{10\!\cdots\!76}a^{13}-\frac{27\!\cdots\!19}{36\!\cdots\!92}a^{12}+\frac{36\!\cdots\!19}{27\!\cdots\!44}a^{11}-\frac{56\!\cdots\!05}{36\!\cdots\!92}a^{10}+\frac{51\!\cdots\!09}{16\!\cdots\!34}a^{9}+\frac{41\!\cdots\!73}{54\!\cdots\!88}a^{8}+\frac{63\!\cdots\!87}{10\!\cdots\!76}a^{7}-\frac{12\!\cdots\!57}{10\!\cdots\!76}a^{6}-\frac{31\!\cdots\!11}{36\!\cdots\!92}a^{5}+\frac{28\!\cdots\!31}{13\!\cdots\!72}a^{4}-\frac{38\!\cdots\!07}{45\!\cdots\!24}a^{3}+\frac{35\!\cdots\!79}{13\!\cdots\!72}a^{2}-\frac{51\!\cdots\!13}{22\!\cdots\!12}a+\frac{25\!\cdots\!25}{33\!\cdots\!68}$, $\frac{29\!\cdots\!27}{10\!\cdots\!76}a^{17}-\frac{41\!\cdots\!85}{27\!\cdots\!44}a^{16}+\frac{39\!\cdots\!61}{54\!\cdots\!88}a^{15}-\frac{29\!\cdots\!67}{10\!\cdots\!76}a^{14}+\frac{70\!\cdots\!13}{10\!\cdots\!76}a^{13}-\frac{12\!\cdots\!59}{10\!\cdots\!76}a^{12}+\frac{36\!\cdots\!09}{28\!\cdots\!89}a^{11}+\frac{13\!\cdots\!33}{36\!\cdots\!92}a^{10}-\frac{10\!\cdots\!65}{30\!\cdots\!16}a^{9}-\frac{15\!\cdots\!41}{54\!\cdots\!88}a^{8}+\frac{18\!\cdots\!77}{10\!\cdots\!76}a^{7}+\frac{47\!\cdots\!29}{10\!\cdots\!76}a^{6}-\frac{47\!\cdots\!99}{10\!\cdots\!76}a^{5}+\frac{40\!\cdots\!25}{27\!\cdots\!44}a^{4}+\frac{21\!\cdots\!19}{67\!\cdots\!36}a^{3}-\frac{19\!\cdots\!53}{15\!\cdots\!08}a^{2}-\frac{51\!\cdots\!05}{75\!\cdots\!04}a+\frac{75\!\cdots\!51}{11\!\cdots\!56}$, $\frac{78\!\cdots\!03}{60\!\cdots\!32}a^{17}-\frac{17\!\cdots\!43}{27\!\cdots\!44}a^{16}+\frac{88\!\cdots\!15}{27\!\cdots\!44}a^{15}-\frac{21\!\cdots\!57}{18\!\cdots\!96}a^{14}+\frac{15\!\cdots\!99}{54\!\cdots\!88}a^{13}-\frac{29\!\cdots\!53}{54\!\cdots\!88}a^{12}+\frac{61\!\cdots\!45}{90\!\cdots\!48}a^{11}-\frac{74\!\cdots\!13}{54\!\cdots\!88}a^{10}-\frac{22\!\cdots\!31}{27\!\cdots\!44}a^{9}-\frac{60\!\cdots\!23}{30\!\cdots\!16}a^{8}+\frac{33\!\cdots\!85}{54\!\cdots\!88}a^{7}+\frac{17\!\cdots\!79}{54\!\cdots\!88}a^{6}-\frac{20\!\cdots\!75}{18\!\cdots\!96}a^{5}+\frac{82\!\cdots\!21}{27\!\cdots\!44}a^{4}+\frac{32\!\cdots\!63}{13\!\cdots\!72}a^{3}+\frac{11\!\cdots\!31}{56\!\cdots\!78}a^{2}+\frac{69\!\cdots\!19}{31\!\cdots\!52}a-\frac{70\!\cdots\!36}{84\!\cdots\!67}$, $\frac{43\!\cdots\!81}{10\!\cdots\!76}a^{17}-\frac{24\!\cdots\!17}{27\!\cdots\!44}a^{16}+\frac{11\!\cdots\!43}{54\!\cdots\!88}a^{15}+\frac{12\!\cdots\!23}{10\!\cdots\!76}a^{14}-\frac{74\!\cdots\!33}{10\!\cdots\!76}a^{13}+\frac{94\!\cdots\!45}{36\!\cdots\!92}a^{12}-\frac{29\!\cdots\!87}{45\!\cdots\!24}a^{11}+\frac{44\!\cdots\!43}{36\!\cdots\!92}a^{10}-\frac{28\!\cdots\!09}{27\!\cdots\!44}a^{9}-\frac{95\!\cdots\!75}{54\!\cdots\!88}a^{8}+\frac{20\!\cdots\!99}{10\!\cdots\!76}a^{7}+\frac{11\!\cdots\!47}{10\!\cdots\!76}a^{6}-\frac{98\!\cdots\!53}{10\!\cdots\!76}a^{5}-\frac{51\!\cdots\!15}{27\!\cdots\!44}a^{4}+\frac{16\!\cdots\!41}{75\!\cdots\!04}a^{3}+\frac{21\!\cdots\!55}{45\!\cdots\!24}a^{2}-\frac{48\!\cdots\!35}{75\!\cdots\!04}a-\frac{47\!\cdots\!01}{33\!\cdots\!68}$, $\frac{33\!\cdots\!07}{90\!\cdots\!48}a^{17}-\frac{92\!\cdots\!33}{27\!\cdots\!44}a^{16}+\frac{42\!\cdots\!03}{22\!\cdots\!12}a^{15}-\frac{74\!\cdots\!77}{90\!\cdots\!48}a^{14}+\frac{91\!\cdots\!79}{33\!\cdots\!68}a^{13}-\frac{30\!\cdots\!75}{45\!\cdots\!24}a^{12}+\frac{37\!\cdots\!15}{30\!\cdots\!16}a^{11}-\frac{45\!\cdots\!37}{27\!\cdots\!44}a^{10}+\frac{31\!\cdots\!59}{30\!\cdots\!16}a^{9}-\frac{29\!\cdots\!03}{11\!\cdots\!56}a^{8}+\frac{81\!\cdots\!65}{27\!\cdots\!44}a^{7}-\frac{19\!\cdots\!69}{22\!\cdots\!12}a^{6}+\frac{82\!\cdots\!15}{45\!\cdots\!24}a^{5}+\frac{43\!\cdots\!43}{27\!\cdots\!44}a^{4}-\frac{90\!\cdots\!99}{45\!\cdots\!24}a^{3}+\frac{56\!\cdots\!77}{75\!\cdots\!04}a^{2}+\frac{77\!\cdots\!11}{16\!\cdots\!34}a-\frac{81\!\cdots\!43}{18\!\cdots\!26}$, $\frac{18\!\cdots\!93}{36\!\cdots\!92}a^{17}-\frac{96\!\cdots\!57}{33\!\cdots\!68}a^{16}+\frac{69\!\cdots\!13}{49\!\cdots\!32}a^{15}-\frac{18\!\cdots\!01}{36\!\cdots\!92}a^{14}+\frac{13\!\cdots\!61}{10\!\cdots\!76}a^{13}-\frac{25\!\cdots\!55}{10\!\cdots\!76}a^{12}+\frac{25\!\cdots\!79}{90\!\cdots\!48}a^{11}-\frac{15\!\cdots\!17}{10\!\cdots\!76}a^{10}-\frac{89\!\cdots\!21}{16\!\cdots\!34}a^{9}-\frac{10\!\cdots\!63}{18\!\cdots\!96}a^{8}+\frac{33\!\cdots\!13}{10\!\cdots\!76}a^{7}+\frac{62\!\cdots\!33}{10\!\cdots\!76}a^{6}-\frac{25\!\cdots\!45}{36\!\cdots\!92}a^{5}+\frac{41\!\cdots\!89}{13\!\cdots\!72}a^{4}+\frac{54\!\cdots\!01}{13\!\cdots\!72}a^{3}-\frac{76\!\cdots\!89}{45\!\cdots\!24}a^{2}-\frac{72\!\cdots\!65}{67\!\cdots\!36}a+\frac{71\!\cdots\!19}{33\!\cdots\!68}$, $\frac{30\!\cdots\!17}{36\!\cdots\!92}a^{17}-\frac{68\!\cdots\!51}{13\!\cdots\!12}a^{16}+\frac{44\!\cdots\!51}{18\!\cdots\!96}a^{15}-\frac{11\!\cdots\!79}{12\!\cdots\!64}a^{14}+\frac{30\!\cdots\!57}{12\!\cdots\!64}a^{13}-\frac{18\!\cdots\!89}{36\!\cdots\!92}a^{12}+\frac{66\!\cdots\!21}{90\!\cdots\!48}a^{11}-\frac{55\!\cdots\!13}{12\!\cdots\!64}a^{10}-\frac{20\!\cdots\!93}{45\!\cdots\!24}a^{9}-\frac{15\!\cdots\!19}{18\!\cdots\!96}a^{8}+\frac{61\!\cdots\!77}{12\!\cdots\!64}a^{7}-\frac{43\!\cdots\!97}{36\!\cdots\!92}a^{6}-\frac{11\!\cdots\!07}{12\!\cdots\!64}a^{5}+\frac{30\!\cdots\!03}{37\!\cdots\!52}a^{4}-\frac{10\!\cdots\!93}{45\!\cdots\!24}a^{3}+\frac{96\!\cdots\!03}{45\!\cdots\!24}a^{2}-\frac{63\!\cdots\!85}{75\!\cdots\!04}a-\frac{30\!\cdots\!07}{11\!\cdots\!56}$, $\frac{65\!\cdots\!47}{18\!\cdots\!96}a^{17}-\frac{60\!\cdots\!03}{27\!\cdots\!44}a^{16}+\frac{30\!\cdots\!41}{27\!\cdots\!44}a^{15}-\frac{79\!\cdots\!15}{18\!\cdots\!96}a^{14}+\frac{64\!\cdots\!33}{54\!\cdots\!88}a^{13}-\frac{13\!\cdots\!19}{54\!\cdots\!88}a^{12}+\frac{34\!\cdots\!69}{90\!\cdots\!48}a^{11}-\frac{16\!\cdots\!67}{54\!\cdots\!88}a^{10}-\frac{28\!\cdots\!27}{27\!\cdots\!44}a^{9}-\frac{33\!\cdots\!27}{90\!\cdots\!48}a^{8}+\frac{12\!\cdots\!39}{54\!\cdots\!88}a^{7}-\frac{61\!\cdots\!63}{54\!\cdots\!88}a^{6}-\frac{68\!\cdots\!57}{18\!\cdots\!96}a^{5}+\frac{12\!\cdots\!09}{27\!\cdots\!44}a^{4}-\frac{16\!\cdots\!65}{13\!\cdots\!72}a^{3}+\frac{98\!\cdots\!68}{28\!\cdots\!89}a^{2}-\frac{20\!\cdots\!93}{33\!\cdots\!68}a+\frac{55\!\cdots\!88}{84\!\cdots\!67}$, $\frac{61\!\cdots\!75}{36\!\cdots\!92}a^{17}-\frac{25\!\cdots\!69}{27\!\cdots\!44}a^{16}+\frac{26\!\cdots\!07}{54\!\cdots\!88}a^{15}-\frac{22\!\cdots\!37}{12\!\cdots\!64}a^{14}+\frac{51\!\cdots\!39}{10\!\cdots\!76}a^{13}-\frac{10\!\cdots\!29}{10\!\cdots\!76}a^{12}+\frac{63\!\cdots\!55}{45\!\cdots\!24}a^{11}-\frac{95\!\cdots\!79}{10\!\cdots\!76}a^{10}-\frac{20\!\cdots\!85}{27\!\cdots\!44}a^{9}-\frac{36\!\cdots\!41}{18\!\cdots\!96}a^{8}+\frac{10\!\cdots\!55}{10\!\cdots\!76}a^{7}-\frac{14\!\cdots\!09}{10\!\cdots\!76}a^{6}-\frac{18\!\cdots\!61}{12\!\cdots\!64}a^{5}+\frac{40\!\cdots\!85}{27\!\cdots\!44}a^{4}-\frac{21\!\cdots\!16}{84\!\cdots\!67}a^{3}+\frac{27\!\cdots\!83}{45\!\cdots\!24}a^{2}-\frac{14\!\cdots\!67}{67\!\cdots\!36}a-\frac{33\!\cdots\!37}{33\!\cdots\!68}$
| 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: | \( 22453155.9774 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 22453155.9774 \cdot 2}{2\cdot\sqrt{11360989554893559098699461641}}\cr\approx \mathstrut & 0.829523124274 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64)
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^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64, 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^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64);
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^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64);
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
$C_3^2:A_4$ (as 18T48):
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 108 |
| The 20 conjugacy class representatives for $C_3^2:A_4$ |
| Character table for $C_3^2:A_4$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 6.2.2368521.1, 9.9.5609891727441.1 |
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 18 siblings: | data not computed |
| Degree 36 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 | ${\href{/padicField/2.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.16.1 | $x^{9} + 3 x^{8} + 3 x^{6} + 3$ | $9$ | $1$ | $16$ | $C_3^2:C_3$ | $[2, 2]^{3}$ |
| 3.9.16.1 | $x^{9} + 3 x^{8} + 3 x^{6} + 3$ | $9$ | $1$ | $16$ | $C_3^2:C_3$ | $[2, 2]^{3}$ | |
|
\(19\)
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.6.5.1 | $x^{6} + 38$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 19.6.5.3 | $x^{6} + 190$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |