Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13)
gp: K = bnfinit(y^18 - y^17 - 15*y^16 + 26*y^15 + 102*y^14 - 154*y^13 - 658*y^12 + 1123*y^11 + 1768*y^10 - 3055*y^9 - 5130*y^8 + 7964*y^7 + 3857*y^6 - 7735*y^5 - 6123*y^4 + 4758*y^3 + 2938*y^2 - 312*y - 13, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13)
\( x^{18} - x^{17} - 15 x^{16} + 26 x^{15} + 102 x^{14} - 154 x^{13} - 658 x^{12} + 1123 x^{11} + 1768 x^{10} + \cdots - 13 \)
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: |
\(774263645280065083180300081\)
\(\medspace = 7^{16}\cdot 13^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.18\) | 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: | $7^{8/9}13^{5/6}\approx 47.806217686235044$ | ||
| Ramified primes: |
\(7\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{33\!\cdots\!93}a^{17}-\frac{81\!\cdots\!90}{33\!\cdots\!93}a^{16}-\frac{86\!\cdots\!73}{33\!\cdots\!93}a^{15}-\frac{71\!\cdots\!19}{33\!\cdots\!93}a^{14}+\frac{10\!\cdots\!09}{33\!\cdots\!93}a^{13}+\frac{11\!\cdots\!59}{33\!\cdots\!93}a^{12}+\frac{29\!\cdots\!79}{33\!\cdots\!93}a^{11}+\frac{32\!\cdots\!19}{33\!\cdots\!93}a^{10}+\frac{75\!\cdots\!92}{33\!\cdots\!93}a^{9}+\frac{15\!\cdots\!85}{33\!\cdots\!93}a^{8}+\frac{76\!\cdots\!69}{33\!\cdots\!93}a^{7}+\frac{12\!\cdots\!63}{33\!\cdots\!93}a^{6}+\frac{90\!\cdots\!42}{33\!\cdots\!93}a^{5}-\frac{14\!\cdots\!06}{33\!\cdots\!93}a^{4}-\frac{77\!\cdots\!31}{33\!\cdots\!93}a^{3}+\frac{36\!\cdots\!91}{33\!\cdots\!93}a^{2}+\frac{66\!\cdots\!76}{33\!\cdots\!93}a-\frac{14\!\cdots\!89}{33\!\cdots\!93}$
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 $1$ (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: |
$\frac{25\!\cdots\!61}{19\!\cdots\!87}a^{17}+\frac{14\!\cdots\!59}{19\!\cdots\!87}a^{16}-\frac{67\!\cdots\!83}{19\!\cdots\!87}a^{15}-\frac{13\!\cdots\!82}{19\!\cdots\!87}a^{14}+\frac{86\!\cdots\!53}{19\!\cdots\!87}a^{13}+\frac{43\!\cdots\!98}{19\!\cdots\!87}a^{12}-\frac{46\!\cdots\!89}{19\!\cdots\!87}a^{11}-\frac{25\!\cdots\!84}{19\!\cdots\!87}a^{10}+\frac{27\!\cdots\!78}{19\!\cdots\!87}a^{9}-\frac{17\!\cdots\!56}{19\!\cdots\!87}a^{8}-\frac{50\!\cdots\!94}{19\!\cdots\!87}a^{7}+\frac{32\!\cdots\!58}{19\!\cdots\!87}a^{6}+\frac{13\!\cdots\!29}{19\!\cdots\!87}a^{5}-\frac{26\!\cdots\!32}{19\!\cdots\!87}a^{4}+\frac{35\!\cdots\!60}{19\!\cdots\!87}a^{3}+\frac{10\!\cdots\!32}{19\!\cdots\!87}a^{2}-\frac{99\!\cdots\!68}{19\!\cdots\!87}a-\frac{15\!\cdots\!45}{19\!\cdots\!87}$, $\frac{22\!\cdots\!91}{19\!\cdots\!87}a^{17}+\frac{27\!\cdots\!10}{19\!\cdots\!87}a^{16}-\frac{82\!\cdots\!20}{19\!\cdots\!87}a^{15}-\frac{31\!\cdots\!62}{19\!\cdots\!87}a^{14}+\frac{12\!\cdots\!43}{19\!\cdots\!87}a^{13}+\frac{13\!\cdots\!45}{19\!\cdots\!87}a^{12}-\frac{65\!\cdots\!74}{19\!\cdots\!87}a^{11}-\frac{80\!\cdots\!17}{19\!\cdots\!87}a^{10}+\frac{41\!\cdots\!86}{19\!\cdots\!87}a^{9}-\frac{13\!\cdots\!99}{19\!\cdots\!87}a^{8}-\frac{76\!\cdots\!73}{19\!\cdots\!87}a^{7}+\frac{32\!\cdots\!06}{19\!\cdots\!87}a^{6}+\frac{17\!\cdots\!33}{19\!\cdots\!87}a^{5}-\frac{31\!\cdots\!46}{19\!\cdots\!87}a^{4}+\frac{36\!\cdots\!76}{19\!\cdots\!87}a^{3}+\frac{13\!\cdots\!39}{19\!\cdots\!87}a^{2}-\frac{11\!\cdots\!76}{19\!\cdots\!87}a+\frac{10\!\cdots\!11}{19\!\cdots\!87}$, $\frac{38\!\cdots\!18}{33\!\cdots\!93}a^{17}-\frac{23\!\cdots\!95}{33\!\cdots\!93}a^{16}-\frac{50\!\cdots\!22}{33\!\cdots\!93}a^{15}+\frac{39\!\cdots\!34}{33\!\cdots\!93}a^{14}+\frac{63\!\cdots\!02}{33\!\cdots\!93}a^{13}-\frac{27\!\cdots\!74}{33\!\cdots\!93}a^{12}-\frac{78\!\cdots\!89}{33\!\cdots\!93}a^{11}+\frac{17\!\cdots\!90}{33\!\cdots\!93}a^{10}-\frac{70\!\cdots\!17}{33\!\cdots\!93}a^{9}-\frac{53\!\cdots\!17}{33\!\cdots\!93}a^{8}+\frac{17\!\cdots\!98}{33\!\cdots\!93}a^{7}+\frac{14\!\cdots\!42}{33\!\cdots\!93}a^{6}-\frac{76\!\cdots\!73}{33\!\cdots\!93}a^{5}-\frac{15\!\cdots\!22}{33\!\cdots\!93}a^{4}+\frac{81\!\cdots\!84}{33\!\cdots\!93}a^{3}+\frac{16\!\cdots\!98}{33\!\cdots\!93}a^{2}-\frac{16\!\cdots\!20}{33\!\cdots\!93}a-\frac{54\!\cdots\!34}{33\!\cdots\!93}$, $\frac{31\!\cdots\!96}{33\!\cdots\!93}a^{17}-\frac{52\!\cdots\!40}{33\!\cdots\!93}a^{16}-\frac{42\!\cdots\!05}{33\!\cdots\!93}a^{15}+\frac{10\!\cdots\!90}{33\!\cdots\!93}a^{14}+\frac{22\!\cdots\!26}{33\!\cdots\!93}a^{13}-\frac{60\!\cdots\!65}{33\!\cdots\!93}a^{12}-\frac{15\!\cdots\!83}{33\!\cdots\!93}a^{11}+\frac{43\!\cdots\!39}{33\!\cdots\!93}a^{10}+\frac{17\!\cdots\!46}{33\!\cdots\!93}a^{9}-\frac{93\!\cdots\!55}{33\!\cdots\!93}a^{8}-\frac{72\!\cdots\!87}{33\!\cdots\!93}a^{7}+\frac{26\!\cdots\!73}{33\!\cdots\!93}a^{6}-\frac{12\!\cdots\!40}{33\!\cdots\!93}a^{5}-\frac{68\!\cdots\!53}{33\!\cdots\!93}a^{4}-\frac{10\!\cdots\!71}{33\!\cdots\!93}a^{3}+\frac{10\!\cdots\!92}{33\!\cdots\!93}a^{2}-\frac{84\!\cdots\!79}{33\!\cdots\!93}a-\frac{13\!\cdots\!04}{33\!\cdots\!93}$, $\frac{21\!\cdots\!73}{33\!\cdots\!93}a^{17}-\frac{19\!\cdots\!24}{33\!\cdots\!93}a^{16}-\frac{32\!\cdots\!54}{33\!\cdots\!93}a^{15}+\frac{52\!\cdots\!71}{33\!\cdots\!93}a^{14}+\frac{22\!\cdots\!45}{33\!\cdots\!93}a^{13}-\frac{30\!\cdots\!53}{33\!\cdots\!93}a^{12}-\frac{14\!\cdots\!72}{33\!\cdots\!93}a^{11}+\frac{22\!\cdots\!89}{33\!\cdots\!93}a^{10}+\frac{39\!\cdots\!53}{33\!\cdots\!93}a^{9}-\frac{60\!\cdots\!49}{33\!\cdots\!93}a^{8}-\frac{11\!\cdots\!60}{33\!\cdots\!93}a^{7}+\frac{15\!\cdots\!71}{33\!\cdots\!93}a^{6}+\frac{96\!\cdots\!53}{33\!\cdots\!93}a^{5}-\frac{15\!\cdots\!92}{33\!\cdots\!93}a^{4}-\frac{14\!\cdots\!29}{33\!\cdots\!93}a^{3}+\frac{82\!\cdots\!45}{33\!\cdots\!93}a^{2}+\frac{71\!\cdots\!88}{33\!\cdots\!93}a+\frac{31\!\cdots\!22}{33\!\cdots\!93}$, $\frac{30\!\cdots\!23}{33\!\cdots\!93}a^{17}-\frac{57\!\cdots\!14}{33\!\cdots\!93}a^{16}-\frac{43\!\cdots\!67}{33\!\cdots\!93}a^{15}+\frac{11\!\cdots\!09}{33\!\cdots\!93}a^{14}+\frac{24\!\cdots\!21}{33\!\cdots\!93}a^{13}-\frac{72\!\cdots\!83}{33\!\cdots\!93}a^{12}-\frac{16\!\cdots\!69}{33\!\cdots\!93}a^{11}+\frac{50\!\cdots\!41}{33\!\cdots\!93}a^{10}+\frac{25\!\cdots\!39}{33\!\cdots\!93}a^{9}-\frac{13\!\cdots\!96}{33\!\cdots\!93}a^{8}-\frac{85\!\cdots\!38}{33\!\cdots\!93}a^{7}+\frac{35\!\cdots\!56}{33\!\cdots\!93}a^{6}-\frac{74\!\cdots\!00}{33\!\cdots\!93}a^{5}-\frac{26\!\cdots\!64}{33\!\cdots\!93}a^{4}-\frac{31\!\cdots\!97}{33\!\cdots\!93}a^{3}+\frac{25\!\cdots\!47}{33\!\cdots\!93}a^{2}-\frac{22\!\cdots\!53}{33\!\cdots\!93}a-\frac{12\!\cdots\!15}{33\!\cdots\!93}$, $\frac{18\!\cdots\!09}{33\!\cdots\!93}a^{17}-\frac{70\!\cdots\!39}{33\!\cdots\!93}a^{16}-\frac{27\!\cdots\!10}{33\!\cdots\!93}a^{15}+\frac{31\!\cdots\!17}{33\!\cdots\!93}a^{14}+\frac{20\!\cdots\!51}{33\!\cdots\!93}a^{13}-\frac{15\!\cdots\!05}{33\!\cdots\!93}a^{12}-\frac{12\!\cdots\!79}{33\!\cdots\!93}a^{11}+\frac{13\!\cdots\!75}{33\!\cdots\!93}a^{10}+\frac{37\!\cdots\!25}{33\!\cdots\!93}a^{9}-\frac{32\!\cdots\!44}{33\!\cdots\!93}a^{8}-\frac{10\!\cdots\!08}{33\!\cdots\!93}a^{7}+\frac{79\!\cdots\!40}{33\!\cdots\!93}a^{6}+\frac{81\!\cdots\!76}{33\!\cdots\!93}a^{5}-\frac{80\!\cdots\!24}{33\!\cdots\!93}a^{4}-\frac{12\!\cdots\!04}{33\!\cdots\!93}a^{3}-\frac{53\!\cdots\!67}{33\!\cdots\!93}a^{2}+\frac{21\!\cdots\!84}{33\!\cdots\!93}a+\frac{14\!\cdots\!75}{33\!\cdots\!93}$, $\frac{30\!\cdots\!26}{33\!\cdots\!93}a^{17}-\frac{22\!\cdots\!21}{33\!\cdots\!93}a^{16}-\frac{44\!\cdots\!50}{33\!\cdots\!93}a^{15}+\frac{67\!\cdots\!92}{33\!\cdots\!93}a^{14}+\frac{29\!\cdots\!34}{33\!\cdots\!93}a^{13}-\frac{34\!\cdots\!75}{33\!\cdots\!93}a^{12}-\frac{18\!\cdots\!89}{33\!\cdots\!93}a^{11}+\frac{27\!\cdots\!33}{33\!\cdots\!93}a^{10}+\frac{44\!\cdots\!76}{33\!\cdots\!93}a^{9}-\frac{62\!\cdots\!37}{33\!\cdots\!93}a^{8}-\frac{12\!\cdots\!28}{33\!\cdots\!93}a^{7}+\frac{16\!\cdots\!47}{33\!\cdots\!93}a^{6}+\frac{38\!\cdots\!96}{33\!\cdots\!93}a^{5}-\frac{12\!\cdots\!37}{33\!\cdots\!93}a^{4}-\frac{15\!\cdots\!36}{33\!\cdots\!93}a^{3}+\frac{42\!\cdots\!28}{33\!\cdots\!93}a^{2}-\frac{17\!\cdots\!50}{33\!\cdots\!93}a-\frac{26\!\cdots\!41}{33\!\cdots\!93}$, $\frac{18\!\cdots\!46}{33\!\cdots\!93}a^{17}-\frac{36\!\cdots\!54}{33\!\cdots\!93}a^{16}-\frac{27\!\cdots\!13}{33\!\cdots\!93}a^{15}+\frac{76\!\cdots\!94}{33\!\cdots\!93}a^{14}+\frac{16\!\cdots\!58}{33\!\cdots\!93}a^{13}-\frac{49\!\cdots\!47}{33\!\cdots\!93}a^{12}-\frac{11\!\cdots\!89}{33\!\cdots\!93}a^{11}+\frac{34\!\cdots\!09}{33\!\cdots\!93}a^{10}+\frac{22\!\cdots\!12}{33\!\cdots\!93}a^{9}-\frac{99\!\cdots\!57}{33\!\cdots\!93}a^{8}-\frac{68\!\cdots\!80}{33\!\cdots\!93}a^{7}+\frac{26\!\cdots\!93}{33\!\cdots\!93}a^{6}-\frac{15\!\cdots\!08}{33\!\cdots\!93}a^{5}-\frac{24\!\cdots\!43}{33\!\cdots\!93}a^{4}-\frac{10\!\cdots\!17}{33\!\cdots\!93}a^{3}+\frac{21\!\cdots\!98}{33\!\cdots\!93}a^{2}-\frac{18\!\cdots\!43}{33\!\cdots\!93}a-\frac{14\!\cdots\!03}{33\!\cdots\!93}$, $\frac{58\!\cdots\!03}{33\!\cdots\!93}a^{17}-\frac{58\!\cdots\!85}{33\!\cdots\!93}a^{16}-\frac{88\!\cdots\!04}{33\!\cdots\!93}a^{15}+\frac{15\!\cdots\!03}{33\!\cdots\!93}a^{14}+\frac{60\!\cdots\!41}{33\!\cdots\!93}a^{13}-\frac{91\!\cdots\!15}{33\!\cdots\!93}a^{12}-\frac{39\!\cdots\!58}{33\!\cdots\!93}a^{11}+\frac{66\!\cdots\!87}{33\!\cdots\!93}a^{10}+\frac{10\!\cdots\!79}{33\!\cdots\!93}a^{9}-\frac{18\!\cdots\!61}{33\!\cdots\!93}a^{8}-\frac{31\!\cdots\!33}{33\!\cdots\!93}a^{7}+\frac{47\!\cdots\!93}{33\!\cdots\!93}a^{6}+\frac{26\!\cdots\!43}{33\!\cdots\!93}a^{5}-\frac{46\!\cdots\!87}{33\!\cdots\!93}a^{4}-\frac{39\!\cdots\!85}{33\!\cdots\!93}a^{3}+\frac{29\!\cdots\!69}{33\!\cdots\!93}a^{2}+\frac{22\!\cdots\!85}{33\!\cdots\!93}a+\frac{66\!\cdots\!71}{33\!\cdots\!93}$, $\frac{46\!\cdots\!90}{33\!\cdots\!93}a^{17}-\frac{47\!\cdots\!55}{33\!\cdots\!93}a^{16}-\frac{70\!\cdots\!87}{33\!\cdots\!93}a^{15}+\frac{12\!\cdots\!65}{33\!\cdots\!93}a^{14}+\frac{47\!\cdots\!67}{33\!\cdots\!93}a^{13}-\frac{73\!\cdots\!62}{33\!\cdots\!93}a^{12}-\frac{30\!\cdots\!08}{33\!\cdots\!93}a^{11}+\frac{53\!\cdots\!69}{33\!\cdots\!93}a^{10}+\frac{81\!\cdots\!75}{33\!\cdots\!93}a^{9}-\frac{14\!\cdots\!07}{33\!\cdots\!93}a^{8}-\frac{23\!\cdots\!68}{33\!\cdots\!93}a^{7}+\frac{38\!\cdots\!06}{33\!\cdots\!93}a^{6}+\frac{17\!\cdots\!14}{33\!\cdots\!93}a^{5}-\frac{37\!\cdots\!76}{33\!\cdots\!93}a^{4}-\frac{27\!\cdots\!26}{33\!\cdots\!93}a^{3}+\frac{23\!\cdots\!17}{33\!\cdots\!93}a^{2}+\frac{13\!\cdots\!46}{33\!\cdots\!93}a-\frac{19\!\cdots\!01}{33\!\cdots\!93}$
| 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: | \( 2770505.95053 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 2770505.95053 \cdot 1}{2\cdot\sqrt{774263645280065083180300081}}\cr\approx \mathstrut & 0.196039615388 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13)
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 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13, 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 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13);
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 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13);
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 18T47):
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
| \(\Q(\zeta_{7})^+\), 6.2.405769.1, 9.9.164648481361.1 |
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 sibling: | data not computed |
| Minimal sibling: | 18.6.4581441688047722385682249.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ | |
|
\(13\)
| 13.6.5.3 | $x^{6} + 39$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |