Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088)
gp: K = bnfinit(y^18 - 36*y^16 - 24*y^15 + 432*y^14 + 576*y^13 - 2088*y^12 - 3888*y^11 + 4284*y^10 + 10240*y^9 - 5040*y^7 - 13530*y^6 - 32832*y^5 + 9576*y^4 + 63360*y^3 + 7128*y^2 - 1152*y - 1088, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088)
\( x^{18} - 36 x^{16} - 24 x^{15} + 432 x^{14} + 576 x^{13} - 2088 x^{12} - 3888 x^{11} + 4284 x^{10} + \cdots - 1088 \)
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: |
\(95057738942888775593520682500096000\)
\(\medspace = 2^{49}\cdot 3^{38}\cdot 5^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(87.74\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{10}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{18}a^{12}+\frac{1}{9}a^{9}-\frac{1}{3}a^{6}+\frac{4}{9}a^{3}+\frac{2}{9}$, $\frac{1}{18}a^{13}+\frac{1}{9}a^{10}-\frac{1}{3}a^{7}+\frac{4}{9}a^{4}+\frac{2}{9}a$, $\frac{1}{36}a^{14}-\frac{1}{9}a^{11}+\frac{1}{3}a^{8}+\frac{2}{9}a^{5}+\frac{5}{18}a^{2}$, $\frac{1}{72}a^{15}-\frac{1}{36}a^{13}-\frac{1}{6}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{6}a^{7}-\frac{2}{9}a^{6}-\frac{2}{9}a^{4}+\frac{1}{12}a^{3}-\frac{1}{3}a^{2}-\frac{5}{18}a-\frac{1}{9}$, $\frac{1}{288}a^{16}+\frac{1}{144}a^{14}-\frac{1}{36}a^{13}-\frac{1}{72}a^{12}+\frac{1}{18}a^{11}-\frac{1}{9}a^{10}+\frac{1}{18}a^{9}+\frac{5}{24}a^{8}+\frac{1}{9}a^{7}-\frac{5}{12}a^{6}+\frac{1}{18}a^{5}+\frac{43}{144}a^{4}-\frac{1}{9}a^{3}+\frac{35}{72}a^{2}+\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{51\!\cdots\!64}a^{17}-\frac{65\!\cdots\!39}{51\!\cdots\!64}a^{16}+\frac{52\!\cdots\!03}{25\!\cdots\!32}a^{15}-\frac{44\!\cdots\!81}{85\!\cdots\!44}a^{14}+\frac{31\!\cdots\!19}{12\!\cdots\!16}a^{13}+\frac{42\!\cdots\!31}{12\!\cdots\!16}a^{12}-\frac{19\!\cdots\!58}{16\!\cdots\!27}a^{11}-\frac{15\!\cdots\!63}{10\!\cdots\!18}a^{10}-\frac{55\!\cdots\!07}{42\!\cdots\!72}a^{9}-\frac{59\!\cdots\!89}{12\!\cdots\!16}a^{8}+\frac{14\!\cdots\!63}{64\!\cdots\!08}a^{7}+\frac{79\!\cdots\!03}{64\!\cdots\!08}a^{6}-\frac{10\!\cdots\!81}{28\!\cdots\!48}a^{5}-\frac{70\!\cdots\!97}{25\!\cdots\!32}a^{4}+\frac{25\!\cdots\!97}{12\!\cdots\!16}a^{3}-\frac{61\!\cdots\!97}{12\!\cdots\!16}a^{2}+\frac{40\!\cdots\!07}{10\!\cdots\!18}a-\frac{16\!\cdots\!68}{17\!\cdots\!03}$
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
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: | $11$ | 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: |
$\frac{49\!\cdots\!70}{17\!\cdots\!03}a^{17}+\frac{90\!\cdots\!10}{53\!\cdots\!09}a^{16}-\frac{15\!\cdots\!48}{16\!\cdots\!27}a^{15}-\frac{21\!\cdots\!37}{17\!\cdots\!03}a^{14}+\frac{18\!\cdots\!21}{17\!\cdots\!03}a^{13}+\frac{11\!\cdots\!97}{53\!\cdots\!09}a^{12}-\frac{68\!\cdots\!14}{17\!\cdots\!03}a^{11}-\frac{59\!\cdots\!06}{53\!\cdots\!09}a^{10}+\frac{62\!\cdots\!62}{16\!\cdots\!27}a^{9}+\frac{40\!\cdots\!76}{17\!\cdots\!03}a^{8}+\frac{57\!\cdots\!97}{53\!\cdots\!09}a^{7}+\frac{42\!\cdots\!77}{16\!\cdots\!27}a^{6}-\frac{46\!\cdots\!92}{17\!\cdots\!03}a^{5}-\frac{16\!\cdots\!00}{17\!\cdots\!03}a^{4}-\frac{28\!\cdots\!15}{17\!\cdots\!03}a^{3}+\frac{20\!\cdots\!40}{17\!\cdots\!03}a^{2}+\frac{63\!\cdots\!22}{53\!\cdots\!09}a+\frac{10\!\cdots\!37}{16\!\cdots\!27}$, $\frac{19\!\cdots\!66}{17\!\cdots\!03}a^{17}-\frac{60\!\cdots\!14}{53\!\cdots\!09}a^{16}-\frac{64\!\cdots\!40}{16\!\cdots\!27}a^{15}-\frac{40\!\cdots\!17}{17\!\cdots\!03}a^{14}+\frac{87\!\cdots\!19}{17\!\cdots\!03}a^{13}+\frac{31\!\cdots\!39}{53\!\cdots\!09}a^{12}-\frac{43\!\cdots\!98}{17\!\cdots\!03}a^{11}-\frac{21\!\cdots\!78}{53\!\cdots\!09}a^{10}+\frac{87\!\cdots\!58}{16\!\cdots\!27}a^{9}+\frac{19\!\cdots\!36}{17\!\cdots\!03}a^{8}-\frac{89\!\cdots\!57}{53\!\cdots\!09}a^{7}-\frac{95\!\cdots\!71}{16\!\cdots\!27}a^{6}-\frac{25\!\cdots\!96}{17\!\cdots\!03}a^{5}-\frac{63\!\cdots\!20}{17\!\cdots\!03}a^{4}+\frac{85\!\cdots\!05}{53\!\cdots\!09}a^{3}+\frac{12\!\cdots\!60}{17\!\cdots\!03}a^{2}-\frac{63\!\cdots\!34}{53\!\cdots\!09}a-\frac{65\!\cdots\!11}{16\!\cdots\!27}$, $\frac{31\!\cdots\!32}{17\!\cdots\!03}a^{17}+\frac{36\!\cdots\!98}{17\!\cdots\!03}a^{16}-\frac{12\!\cdots\!97}{21\!\cdots\!36}a^{15}-\frac{90\!\cdots\!44}{17\!\cdots\!03}a^{14}+\frac{22\!\cdots\!11}{35\!\cdots\!06}a^{13}+\frac{16\!\cdots\!19}{16\!\cdots\!27}a^{12}-\frac{40\!\cdots\!91}{17\!\cdots\!03}a^{11}-\frac{29\!\cdots\!00}{53\!\cdots\!09}a^{10}+\frac{35\!\cdots\!35}{16\!\cdots\!27}a^{9}+\frac{17\!\cdots\!32}{17\!\cdots\!03}a^{8}+\frac{13\!\cdots\!66}{17\!\cdots\!03}a^{7}+\frac{24\!\cdots\!48}{53\!\cdots\!09}a^{6}-\frac{29\!\cdots\!16}{17\!\cdots\!03}a^{5}-\frac{91\!\cdots\!52}{17\!\cdots\!03}a^{4}-\frac{12\!\cdots\!35}{32\!\cdots\!54}a^{3}+\frac{75\!\cdots\!92}{17\!\cdots\!03}a^{2}+\frac{77\!\cdots\!37}{53\!\cdots\!09}a+\frac{50\!\cdots\!53}{16\!\cdots\!27}$, $\frac{81\!\cdots\!47}{51\!\cdots\!64}a^{17}+\frac{16\!\cdots\!19}{17\!\cdots\!88}a^{16}-\frac{14\!\cdots\!59}{25\!\cdots\!32}a^{15}-\frac{18\!\cdots\!31}{25\!\cdots\!32}a^{14}+\frac{26\!\cdots\!83}{42\!\cdots\!72}a^{13}+\frac{15\!\cdots\!83}{12\!\cdots\!16}a^{12}-\frac{43\!\cdots\!81}{17\!\cdots\!03}a^{11}-\frac{73\!\cdots\!23}{10\!\cdots\!18}a^{10}+\frac{37\!\cdots\!53}{12\!\cdots\!16}a^{9}+\frac{19\!\cdots\!63}{12\!\cdots\!16}a^{8}+\frac{13\!\cdots\!99}{21\!\cdots\!36}a^{7}-\frac{12\!\cdots\!41}{64\!\cdots\!08}a^{6}-\frac{43\!\cdots\!03}{25\!\cdots\!32}a^{5}-\frac{46\!\cdots\!11}{85\!\cdots\!44}a^{4}-\frac{15\!\cdots\!77}{12\!\cdots\!16}a^{3}+\frac{33\!\cdots\!33}{42\!\cdots\!72}a^{2}+\frac{30\!\cdots\!65}{10\!\cdots\!18}a+\frac{90\!\cdots\!46}{16\!\cdots\!27}$, $\frac{36\!\cdots\!77}{12\!\cdots\!16}a^{17}-\frac{73\!\cdots\!23}{16\!\cdots\!27}a^{16}-\frac{65\!\cdots\!01}{64\!\cdots\!08}a^{15}+\frac{15\!\cdots\!44}{16\!\cdots\!27}a^{14}+\frac{21\!\cdots\!64}{16\!\cdots\!27}a^{13}-\frac{51\!\cdots\!74}{16\!\cdots\!27}a^{12}-\frac{14\!\cdots\!94}{16\!\cdots\!27}a^{11}-\frac{31\!\cdots\!55}{16\!\cdots\!27}a^{10}+\frac{98\!\cdots\!79}{32\!\cdots\!54}a^{9}+\frac{19\!\cdots\!75}{16\!\cdots\!27}a^{8}-\frac{76\!\cdots\!51}{16\!\cdots\!27}a^{7}-\frac{30\!\cdots\!17}{16\!\cdots\!27}a^{6}-\frac{11\!\cdots\!17}{64\!\cdots\!08}a^{5}-\frac{52\!\cdots\!99}{16\!\cdots\!27}a^{4}+\frac{58\!\cdots\!07}{32\!\cdots\!54}a^{3}+\frac{24\!\cdots\!62}{16\!\cdots\!27}a^{2}-\frac{41\!\cdots\!90}{16\!\cdots\!27}a-\frac{86\!\cdots\!27}{16\!\cdots\!27}$, $\frac{98\!\cdots\!87}{42\!\cdots\!72}a^{17}+\frac{46\!\cdots\!97}{51\!\cdots\!64}a^{16}-\frac{10\!\cdots\!79}{12\!\cdots\!16}a^{15}-\frac{22\!\cdots\!31}{25\!\cdots\!32}a^{14}+\frac{31\!\cdots\!71}{32\!\cdots\!54}a^{13}+\frac{22\!\cdots\!55}{12\!\cdots\!16}a^{12}-\frac{72\!\cdots\!68}{16\!\cdots\!27}a^{11}-\frac{18\!\cdots\!64}{16\!\cdots\!27}a^{10}+\frac{11\!\cdots\!69}{16\!\cdots\!27}a^{9}+\frac{42\!\cdots\!91}{14\!\cdots\!24}a^{8}+\frac{23\!\cdots\!05}{32\!\cdots\!54}a^{7}-\frac{11\!\cdots\!95}{64\!\cdots\!08}a^{6}-\frac{23\!\cdots\!59}{64\!\cdots\!08}a^{5}-\frac{22\!\cdots\!05}{25\!\cdots\!32}a^{4}-\frac{18\!\cdots\!37}{64\!\cdots\!08}a^{3}+\frac{22\!\cdots\!71}{12\!\cdots\!16}a^{2}+\frac{23\!\cdots\!75}{32\!\cdots\!54}a-\frac{45\!\cdots\!14}{16\!\cdots\!27}$, $\frac{70\!\cdots\!53}{17\!\cdots\!88}a^{17}-\frac{35\!\cdots\!69}{51\!\cdots\!64}a^{16}-\frac{43\!\cdots\!19}{25\!\cdots\!32}a^{15}+\frac{93\!\cdots\!33}{85\!\cdots\!44}a^{14}+\frac{33\!\cdots\!13}{12\!\cdots\!16}a^{13}+\frac{15\!\cdots\!69}{12\!\cdots\!16}a^{12}-\frac{32\!\cdots\!68}{17\!\cdots\!03}a^{11}-\frac{90\!\cdots\!57}{35\!\cdots\!06}a^{10}+\frac{56\!\cdots\!85}{12\!\cdots\!16}a^{9}+\frac{44\!\cdots\!19}{42\!\cdots\!72}a^{8}+\frac{19\!\cdots\!97}{64\!\cdots\!08}a^{7}-\frac{38\!\cdots\!75}{64\!\cdots\!08}a^{6}-\frac{15\!\cdots\!65}{85\!\cdots\!44}a^{5}-\frac{86\!\cdots\!67}{25\!\cdots\!32}a^{4}-\frac{30\!\cdots\!25}{12\!\cdots\!16}a^{3}+\frac{63\!\cdots\!65}{14\!\cdots\!24}a^{2}+\frac{17\!\cdots\!71}{10\!\cdots\!18}a+\frac{50\!\cdots\!10}{16\!\cdots\!27}$, $\frac{85\!\cdots\!13}{16\!\cdots\!27}a^{17}+\frac{89\!\cdots\!11}{51\!\cdots\!64}a^{16}-\frac{24\!\cdots\!27}{12\!\cdots\!16}a^{15}-\frac{48\!\cdots\!65}{25\!\cdots\!32}a^{14}+\frac{73\!\cdots\!65}{32\!\cdots\!54}a^{13}+\frac{49\!\cdots\!25}{12\!\cdots\!16}a^{12}-\frac{56\!\cdots\!03}{53\!\cdots\!09}a^{11}-\frac{39\!\cdots\!89}{16\!\cdots\!27}a^{10}+\frac{57\!\cdots\!23}{32\!\cdots\!54}a^{9}+\frac{84\!\cdots\!97}{12\!\cdots\!16}a^{8}+\frac{43\!\cdots\!21}{32\!\cdots\!54}a^{7}-\frac{23\!\cdots\!09}{64\!\cdots\!08}a^{6}-\frac{25\!\cdots\!93}{32\!\cdots\!54}a^{5}-\frac{49\!\cdots\!11}{25\!\cdots\!32}a^{4}+\frac{23\!\cdots\!43}{64\!\cdots\!08}a^{3}+\frac{54\!\cdots\!29}{14\!\cdots\!24}a^{2}+\frac{44\!\cdots\!63}{32\!\cdots\!54}a-\frac{91\!\cdots\!92}{16\!\cdots\!27}$, $\frac{30\!\cdots\!95}{64\!\cdots\!08}a^{17}+\frac{13\!\cdots\!73}{25\!\cdots\!32}a^{16}-\frac{26\!\cdots\!69}{16\!\cdots\!27}a^{15}-\frac{37\!\cdots\!15}{12\!\cdots\!16}a^{14}+\frac{29\!\cdots\!94}{16\!\cdots\!27}a^{13}+\frac{30\!\cdots\!23}{64\!\cdots\!08}a^{12}-\frac{97\!\cdots\!28}{16\!\cdots\!27}a^{11}-\frac{43\!\cdots\!90}{16\!\cdots\!27}a^{10}-\frac{40\!\cdots\!02}{16\!\cdots\!27}a^{9}+\frac{39\!\cdots\!95}{64\!\cdots\!08}a^{8}+\frac{88\!\cdots\!97}{16\!\cdots\!27}a^{7}-\frac{24\!\cdots\!57}{32\!\cdots\!54}a^{6}-\frac{26\!\cdots\!87}{32\!\cdots\!54}a^{5}-\frac{28\!\cdots\!49}{12\!\cdots\!16}a^{4}-\frac{22\!\cdots\!10}{16\!\cdots\!27}a^{3}+\frac{18\!\cdots\!87}{64\!\cdots\!08}a^{2}+\frac{54\!\cdots\!00}{16\!\cdots\!27}a+\frac{16\!\cdots\!89}{16\!\cdots\!27}$, $\frac{14\!\cdots\!19}{17\!\cdots\!88}a^{17}+\frac{17\!\cdots\!57}{12\!\cdots\!16}a^{16}-\frac{69\!\cdots\!99}{25\!\cdots\!32}a^{15}-\frac{11\!\cdots\!33}{17\!\cdots\!03}a^{14}+\frac{29\!\cdots\!71}{12\!\cdots\!16}a^{13}+\frac{13\!\cdots\!61}{16\!\cdots\!27}a^{12}-\frac{30\!\cdots\!56}{17\!\cdots\!03}a^{11}-\frac{90\!\cdots\!83}{32\!\cdots\!54}a^{10}-\frac{18\!\cdots\!49}{12\!\cdots\!16}a^{9}+\frac{33\!\cdots\!33}{10\!\cdots\!18}a^{8}+\frac{22\!\cdots\!97}{64\!\cdots\!08}a^{7}+\frac{14\!\cdots\!17}{32\!\cdots\!54}a^{6}+\frac{19\!\cdots\!59}{28\!\cdots\!48}a^{5}-\frac{12\!\cdots\!09}{64\!\cdots\!08}a^{4}-\frac{30\!\cdots\!13}{12\!\cdots\!16}a^{3}+\frac{33\!\cdots\!25}{35\!\cdots\!06}a^{2}-\frac{46\!\cdots\!32}{16\!\cdots\!27}a+\frac{82\!\cdots\!87}{16\!\cdots\!27}$, $\frac{47\!\cdots\!49}{53\!\cdots\!09}a^{17}+\frac{71\!\cdots\!29}{56\!\cdots\!96}a^{16}-\frac{11\!\cdots\!61}{42\!\cdots\!72}a^{15}-\frac{53\!\cdots\!09}{85\!\cdots\!44}a^{14}+\frac{39\!\cdots\!04}{17\!\cdots\!03}a^{13}+\frac{35\!\cdots\!69}{42\!\cdots\!72}a^{12}+\frac{17\!\cdots\!88}{17\!\cdots\!03}a^{11}-\frac{53\!\cdots\!96}{17\!\cdots\!03}a^{10}-\frac{39\!\cdots\!97}{10\!\cdots\!18}a^{9}+\frac{82\!\cdots\!77}{42\!\cdots\!72}a^{8}+\frac{31\!\cdots\!81}{35\!\cdots\!06}a^{7}+\frac{20\!\cdots\!83}{21\!\cdots\!36}a^{6}+\frac{41\!\cdots\!71}{10\!\cdots\!18}a^{5}-\frac{43\!\cdots\!93}{28\!\cdots\!48}a^{4}-\frac{90\!\cdots\!03}{21\!\cdots\!36}a^{3}-\frac{18\!\cdots\!65}{14\!\cdots\!24}a^{2}-\frac{84\!\cdots\!49}{35\!\cdots\!06}a-\frac{37\!\cdots\!02}{53\!\cdots\!09}$
| 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: | \( 484266827745 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 484266827745 \cdot 1}{2\cdot\sqrt{95057738942888775593520682500096000}}\cr\approx \mathstrut & 3.09257449197299 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088)
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 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088, 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 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088);
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 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088);
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
$C_3^6:(C_2^3:S_4)$ (as 18T822):
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 139968 |
| The 93 conjugacy class representatives for $C_3^6:(C_2^3:S_4)$ |
| Character table for $C_3^6:(C_2^3:S_4)$ |
Intermediate fields
| 3.1.216.1, 6.2.11943936.4 |
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 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 27 sibling: | 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 | R | R | R | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.11.11 | $x^{4} + 10$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
| 2.12.35.980 | $x^{12} + 12 x^{6} + 8 x^{3} + 10$ | $12$ | $1$ | $35$ | 12T28 | $[2, 3, 4]_{3}^{2}$ | |
|
\(3\)
| 3.9.19.12 | $x^{9} + 9 x^{6} + 18 x^{5} + 6 x^{3} + 9 x^{2} + 21$ | $9$ | $1$ | $19$ | $C_3 \wr S_3 $ | $[3/2, 2, 5/2, 8/3]_{2}$ |
| 3.9.19.28 | $x^{9} + 9 x^{6} + 18 x^{5} + 9 x^{4} + 15 x^{3} + 9 x^{2} + 3$ | $9$ | $1$ | $19$ | $C_3 \wr S_3 $ | $[3/2, 2, 5/2, 8/3]_{2}$ | |
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |