Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151)
gp: K = bnfinit(y^18 - 2*y^17 - 9*y^16 + 46*y^15 - 16*y^14 - 240*y^13 + 381*y^12 + 852*y^11 - 2655*y^10 + 830*y^9 + 5689*y^8 - 2214*y^7 - 2581*y^6 + 822*y^5 + 15360*y^4 + 30650*y^3 + 18846*y^2 + 3312*y - 151, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151)
\( x^{18} - 2 x^{17} - 9 x^{16} + 46 x^{15} - 16 x^{14} - 240 x^{13} + 381 x^{12} + 852 x^{11} - 2655 x^{10} + \cdots - 151 \)
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: |
\(297066099785564011102208000000\)
\(\medspace = 2^{33}\cdot 5^{6}\cdot 19^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.39\) | 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: | $2^{2}5^{1/2}19^{2/3}\approx 63.686501757102$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41}a^{16}-\frac{10}{41}a^{14}-\frac{15}{41}a^{13}+\frac{5}{41}a^{12}-\frac{10}{41}a^{11}-\frac{13}{41}a^{10}+\frac{16}{41}a^{9}+\frac{14}{41}a^{8}-\frac{19}{41}a^{7}+\frac{20}{41}a^{6}+\frac{18}{41}a^{5}+\frac{18}{41}a^{4}+\frac{20}{41}a^{3}+\frac{7}{41}a^{2}+\frac{17}{41}a+\frac{13}{41}$, $\frac{1}{64\!\cdots\!39}a^{17}-\frac{16\!\cdots\!49}{20\!\cdots\!69}a^{16}+\frac{24\!\cdots\!97}{64\!\cdots\!39}a^{15}+\frac{72\!\cdots\!20}{64\!\cdots\!39}a^{14}+\frac{23\!\cdots\!38}{64\!\cdots\!39}a^{13}+\frac{28\!\cdots\!35}{64\!\cdots\!39}a^{12}-\frac{30\!\cdots\!56}{64\!\cdots\!39}a^{11}+\frac{21\!\cdots\!38}{64\!\cdots\!39}a^{10}-\frac{70\!\cdots\!82}{64\!\cdots\!39}a^{9}+\frac{79\!\cdots\!40}{64\!\cdots\!39}a^{8}+\frac{95\!\cdots\!20}{64\!\cdots\!39}a^{7}-\frac{10\!\cdots\!67}{64\!\cdots\!39}a^{6}-\frac{55\!\cdots\!74}{15\!\cdots\!79}a^{5}-\frac{13\!\cdots\!65}{64\!\cdots\!39}a^{4}+\frac{11\!\cdots\!20}{64\!\cdots\!39}a^{3}-\frac{15\!\cdots\!11}{64\!\cdots\!39}a^{2}+\frac{16\!\cdots\!86}{64\!\cdots\!39}a-\frac{84\!\cdots\!30}{64\!\cdots\!39}$
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: |
$\frac{61\!\cdots\!47}{64\!\cdots\!39}a^{17}-\frac{49\!\cdots\!12}{20\!\cdots\!69}a^{16}-\frac{47\!\cdots\!55}{64\!\cdots\!39}a^{15}+\frac{30\!\cdots\!57}{64\!\cdots\!39}a^{14}-\frac{25\!\cdots\!77}{64\!\cdots\!39}a^{13}-\frac{13\!\cdots\!75}{64\!\cdots\!39}a^{12}+\frac{30\!\cdots\!13}{64\!\cdots\!39}a^{11}+\frac{36\!\cdots\!58}{64\!\cdots\!39}a^{10}-\frac{18\!\cdots\!77}{64\!\cdots\!39}a^{9}+\frac{14\!\cdots\!37}{64\!\cdots\!39}a^{8}+\frac{27\!\cdots\!31}{64\!\cdots\!39}a^{7}-\frac{27\!\cdots\!60}{64\!\cdots\!39}a^{6}-\frac{34\!\cdots\!85}{15\!\cdots\!79}a^{5}+\frac{56\!\cdots\!30}{64\!\cdots\!39}a^{4}+\frac{91\!\cdots\!64}{64\!\cdots\!39}a^{3}+\frac{14\!\cdots\!80}{64\!\cdots\!39}a^{2}+\frac{44\!\cdots\!10}{64\!\cdots\!39}a-\frac{11\!\cdots\!25}{64\!\cdots\!39}$, $\frac{19\!\cdots\!47}{15\!\cdots\!79}a^{17}-\frac{15\!\cdots\!65}{50\!\cdots\!09}a^{16}-\frac{15\!\cdots\!81}{15\!\cdots\!79}a^{15}+\frac{98\!\cdots\!93}{15\!\cdots\!79}a^{14}-\frac{79\!\cdots\!06}{15\!\cdots\!79}a^{13}-\frac{43\!\cdots\!68}{15\!\cdots\!79}a^{12}+\frac{96\!\cdots\!00}{15\!\cdots\!79}a^{11}+\frac{12\!\cdots\!61}{15\!\cdots\!79}a^{10}-\frac{58\!\cdots\!86}{15\!\cdots\!79}a^{9}+\frac{44\!\cdots\!44}{15\!\cdots\!79}a^{8}+\frac{90\!\cdots\!84}{15\!\cdots\!79}a^{7}-\frac{87\!\cdots\!28}{15\!\cdots\!79}a^{6}-\frac{81\!\cdots\!22}{15\!\cdots\!79}a^{5}+\frac{19\!\cdots\!36}{15\!\cdots\!79}a^{4}+\frac{29\!\cdots\!63}{15\!\cdots\!79}a^{3}+\frac{46\!\cdots\!60}{15\!\cdots\!79}a^{2}+\frac{14\!\cdots\!63}{15\!\cdots\!79}a-\frac{40\!\cdots\!23}{15\!\cdots\!79}$, $\frac{39\!\cdots\!20}{15\!\cdots\!79}a^{17}-\frac{32\!\cdots\!18}{50\!\cdots\!09}a^{16}-\frac{30\!\cdots\!89}{15\!\cdots\!79}a^{15}+\frac{19\!\cdots\!63}{15\!\cdots\!79}a^{14}-\frac{16\!\cdots\!64}{15\!\cdots\!79}a^{13}-\frac{86\!\cdots\!11}{15\!\cdots\!79}a^{12}+\frac{19\!\cdots\!02}{15\!\cdots\!79}a^{11}+\frac{23\!\cdots\!48}{15\!\cdots\!79}a^{10}-\frac{11\!\cdots\!08}{15\!\cdots\!79}a^{9}+\frac{92\!\cdots\!00}{15\!\cdots\!79}a^{8}+\frac{17\!\cdots\!84}{15\!\cdots\!79}a^{7}-\frac{17\!\cdots\!89}{15\!\cdots\!79}a^{6}-\frac{10\!\cdots\!26}{15\!\cdots\!79}a^{5}+\frac{35\!\cdots\!71}{15\!\cdots\!79}a^{4}+\frac{59\!\cdots\!84}{15\!\cdots\!79}a^{3}+\frac{91\!\cdots\!22}{15\!\cdots\!79}a^{2}+\frac{29\!\cdots\!05}{15\!\cdots\!79}a-\frac{44\!\cdots\!88}{15\!\cdots\!79}$, $\frac{56\!\cdots\!62}{64\!\cdots\!39}a^{17}-\frac{46\!\cdots\!12}{20\!\cdots\!69}a^{16}-\frac{42\!\cdots\!49}{64\!\cdots\!39}a^{15}+\frac{28\!\cdots\!05}{64\!\cdots\!39}a^{14}-\frac{24\!\cdots\!50}{64\!\cdots\!39}a^{13}-\frac{12\!\cdots\!34}{64\!\cdots\!39}a^{12}+\frac{27\!\cdots\!96}{64\!\cdots\!39}a^{11}+\frac{32\!\cdots\!16}{64\!\cdots\!39}a^{10}-\frac{16\!\cdots\!62}{64\!\cdots\!39}a^{9}+\frac{13\!\cdots\!28}{64\!\cdots\!39}a^{8}+\frac{23\!\cdots\!42}{64\!\cdots\!39}a^{7}-\frac{25\!\cdots\!61}{64\!\cdots\!39}a^{6}-\frac{46\!\cdots\!48}{15\!\cdots\!79}a^{5}+\frac{39\!\cdots\!62}{64\!\cdots\!39}a^{4}+\frac{84\!\cdots\!64}{64\!\cdots\!39}a^{3}+\frac{12\!\cdots\!86}{64\!\cdots\!39}a^{2}+\frac{38\!\cdots\!51}{64\!\cdots\!39}a-\frac{86\!\cdots\!58}{64\!\cdots\!39}$, $\frac{26\!\cdots\!59}{41\!\cdots\!87}a^{17}-\frac{17\!\cdots\!72}{13\!\cdots\!77}a^{16}-\frac{23\!\cdots\!04}{41\!\cdots\!87}a^{15}+\frac{12\!\cdots\!00}{41\!\cdots\!87}a^{14}-\frac{54\!\cdots\!56}{41\!\cdots\!87}a^{13}-\frac{61\!\cdots\!09}{41\!\cdots\!87}a^{12}+\frac{10\!\cdots\!57}{41\!\cdots\!87}a^{11}+\frac{21\!\cdots\!02}{41\!\cdots\!87}a^{10}-\frac{70\!\cdots\!43}{41\!\cdots\!87}a^{9}+\frac{27\!\cdots\!49}{41\!\cdots\!87}a^{8}+\frac{13\!\cdots\!63}{41\!\cdots\!87}a^{7}-\frac{52\!\cdots\!29}{41\!\cdots\!87}a^{6}-\frac{67\!\cdots\!72}{41\!\cdots\!87}a^{5}+\frac{14\!\cdots\!40}{41\!\cdots\!87}a^{4}+\frac{39\!\cdots\!28}{41\!\cdots\!87}a^{3}+\frac{78\!\cdots\!50}{41\!\cdots\!87}a^{2}+\frac{49\!\cdots\!76}{41\!\cdots\!87}a-\frac{21\!\cdots\!07}{41\!\cdots\!87}$, $\frac{11\!\cdots\!57}{64\!\cdots\!39}a^{17}-\frac{90\!\cdots\!86}{20\!\cdots\!69}a^{16}-\frac{85\!\cdots\!55}{64\!\cdots\!39}a^{15}+\frac{55\!\cdots\!89}{64\!\cdots\!39}a^{14}-\frac{46\!\cdots\!97}{64\!\cdots\!39}a^{13}-\frac{24\!\cdots\!60}{64\!\cdots\!39}a^{12}+\frac{54\!\cdots\!96}{64\!\cdots\!39}a^{11}+\frac{66\!\cdots\!22}{64\!\cdots\!39}a^{10}-\frac{32\!\cdots\!43}{64\!\cdots\!39}a^{9}+\frac{26\!\cdots\!93}{64\!\cdots\!39}a^{8}+\frac{49\!\cdots\!84}{64\!\cdots\!39}a^{7}-\frac{49\!\cdots\!21}{64\!\cdots\!39}a^{6}-\frac{62\!\cdots\!32}{15\!\cdots\!79}a^{5}+\frac{10\!\cdots\!52}{64\!\cdots\!39}a^{4}+\frac{16\!\cdots\!49}{64\!\cdots\!39}a^{3}+\frac{25\!\cdots\!88}{64\!\cdots\!39}a^{2}+\frac{80\!\cdots\!16}{64\!\cdots\!39}a-\frac{37\!\cdots\!69}{64\!\cdots\!39}$, $\frac{70\!\cdots\!83}{64\!\cdots\!39}a^{17}-\frac{57\!\cdots\!73}{20\!\cdots\!69}a^{16}-\frac{52\!\cdots\!12}{64\!\cdots\!39}a^{15}+\frac{35\!\cdots\!05}{64\!\cdots\!39}a^{14}-\frac{30\!\cdots\!65}{64\!\cdots\!39}a^{13}-\frac{15\!\cdots\!16}{64\!\cdots\!39}a^{12}+\frac{35\!\cdots\!83}{64\!\cdots\!39}a^{11}+\frac{40\!\cdots\!85}{64\!\cdots\!39}a^{10}-\frac{20\!\cdots\!24}{64\!\cdots\!39}a^{9}+\frac{17\!\cdots\!68}{64\!\cdots\!39}a^{8}+\frac{30\!\cdots\!43}{64\!\cdots\!39}a^{7}-\frac{31\!\cdots\!81}{64\!\cdots\!39}a^{6}-\frac{24\!\cdots\!58}{15\!\cdots\!79}a^{5}+\frac{62\!\cdots\!81}{64\!\cdots\!39}a^{4}+\frac{10\!\cdots\!53}{64\!\cdots\!39}a^{3}+\frac{15\!\cdots\!31}{64\!\cdots\!39}a^{2}+\frac{45\!\cdots\!24}{64\!\cdots\!39}a-\frac{18\!\cdots\!58}{64\!\cdots\!39}$, $\frac{22\!\cdots\!80}{64\!\cdots\!39}a^{17}-\frac{19\!\cdots\!71}{20\!\cdots\!69}a^{16}-\frac{15\!\cdots\!87}{64\!\cdots\!39}a^{15}+\frac{11\!\cdots\!84}{64\!\cdots\!39}a^{14}-\frac{11\!\cdots\!24}{64\!\cdots\!39}a^{13}-\frac{46\!\cdots\!51}{64\!\cdots\!39}a^{12}+\frac{12\!\cdots\!00}{64\!\cdots\!39}a^{11}+\frac{10\!\cdots\!47}{64\!\cdots\!39}a^{10}-\frac{69\!\cdots\!04}{64\!\cdots\!39}a^{9}+\frac{68\!\cdots\!76}{64\!\cdots\!39}a^{8}+\frac{90\!\cdots\!09}{64\!\cdots\!39}a^{7}-\frac{13\!\cdots\!74}{64\!\cdots\!39}a^{6}+\frac{39\!\cdots\!14}{15\!\cdots\!79}a^{5}+\frac{31\!\cdots\!05}{64\!\cdots\!39}a^{4}+\frac{32\!\cdots\!30}{64\!\cdots\!39}a^{3}+\frac{43\!\cdots\!20}{64\!\cdots\!39}a^{2}+\frac{43\!\cdots\!08}{64\!\cdots\!39}a-\frac{72\!\cdots\!53}{64\!\cdots\!39}$, $\frac{74\!\cdots\!72}{64\!\cdots\!39}a^{17}-\frac{60\!\cdots\!06}{20\!\cdots\!69}a^{16}-\frac{57\!\cdots\!95}{64\!\cdots\!39}a^{15}+\frac{37\!\cdots\!31}{64\!\cdots\!39}a^{14}-\frac{31\!\cdots\!70}{64\!\cdots\!39}a^{13}-\frac{16\!\cdots\!73}{64\!\cdots\!39}a^{12}+\frac{36\!\cdots\!23}{64\!\cdots\!39}a^{11}+\frac{44\!\cdots\!26}{64\!\cdots\!39}a^{10}-\frac{22\!\cdots\!32}{64\!\cdots\!39}a^{9}+\frac{17\!\cdots\!37}{64\!\cdots\!39}a^{8}+\frac{32\!\cdots\!87}{64\!\cdots\!39}a^{7}-\frac{33\!\cdots\!12}{64\!\cdots\!39}a^{6}-\frac{49\!\cdots\!29}{15\!\cdots\!79}a^{5}+\frac{74\!\cdots\!47}{64\!\cdots\!39}a^{4}+\frac{11\!\cdots\!85}{64\!\cdots\!39}a^{3}+\frac{17\!\cdots\!82}{64\!\cdots\!39}a^{2}+\frac{51\!\cdots\!14}{64\!\cdots\!39}a-\frac{26\!\cdots\!69}{64\!\cdots\!39}$, $\frac{17\!\cdots\!71}{64\!\cdots\!39}a^{17}-\frac{10\!\cdots\!11}{20\!\cdots\!69}a^{16}-\frac{16\!\cdots\!85}{64\!\cdots\!39}a^{15}+\frac{76\!\cdots\!26}{64\!\cdots\!39}a^{14}-\frac{13\!\cdots\!46}{64\!\cdots\!39}a^{13}-\frac{41\!\cdots\!32}{64\!\cdots\!39}a^{12}+\frac{57\!\cdots\!12}{64\!\cdots\!39}a^{11}+\frac{15\!\cdots\!41}{64\!\cdots\!39}a^{10}-\frac{43\!\cdots\!58}{64\!\cdots\!39}a^{9}+\frac{55\!\cdots\!78}{64\!\cdots\!39}a^{8}+\frac{10\!\cdots\!97}{64\!\cdots\!39}a^{7}-\frac{18\!\cdots\!02}{64\!\cdots\!39}a^{6}-\frac{13\!\cdots\!71}{15\!\cdots\!79}a^{5}+\frac{58\!\cdots\!33}{64\!\cdots\!39}a^{4}+\frac{26\!\cdots\!29}{64\!\cdots\!39}a^{3}+\frac{56\!\cdots\!42}{64\!\cdots\!39}a^{2}+\frac{40\!\cdots\!30}{64\!\cdots\!39}a+\frac{92\!\cdots\!04}{64\!\cdots\!39}$, $\frac{51\!\cdots\!94}{64\!\cdots\!39}a^{17}-\frac{26\!\cdots\!50}{20\!\cdots\!69}a^{16}-\frac{45\!\cdots\!28}{64\!\cdots\!39}a^{15}+\frac{21\!\cdots\!14}{64\!\cdots\!39}a^{14}-\frac{91\!\cdots\!09}{64\!\cdots\!39}a^{13}-\frac{11\!\cdots\!43}{64\!\cdots\!39}a^{12}+\frac{23\!\cdots\!43}{64\!\cdots\!39}a^{11}+\frac{34\!\cdots\!77}{64\!\cdots\!39}a^{10}-\frac{15\!\cdots\!61}{64\!\cdots\!39}a^{9}+\frac{13\!\cdots\!73}{64\!\cdots\!39}a^{8}+\frac{39\!\cdots\!53}{64\!\cdots\!39}a^{7}-\frac{67\!\cdots\!16}{64\!\cdots\!39}a^{6}-\frac{11\!\cdots\!73}{15\!\cdots\!79}a^{5}+\frac{14\!\cdots\!12}{64\!\cdots\!39}a^{4}-\frac{21\!\cdots\!63}{64\!\cdots\!39}a^{3}-\frac{14\!\cdots\!88}{64\!\cdots\!39}a^{2}-\frac{47\!\cdots\!61}{64\!\cdots\!39}a+\frac{11\!\cdots\!39}{64\!\cdots\!39}$
| 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: | \( 50327821.72401078 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 50327821.72401078 \cdot 1}{2\cdot\sqrt{297066099785564011102208000000}}\cr\approx \mathstrut & 0.181807073048013 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151)
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 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151, 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 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151);
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 - 2*x^17 - 9*x^16 + 46*x^15 - 16*x^14 - 240*x^13 + 381*x^12 + 852*x^11 - 2655*x^10 + 830*x^9 + 5689*x^8 - 2214*x^7 - 2581*x^6 + 822*x^5 + 15360*x^4 + 30650*x^3 + 18846*x^2 + 3312*x - 151);
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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.3.361.1, 3.1.14440.1, 6.2.6672435200.7, 6.6.66724352.1, 9.3.3010936384000.1 |
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
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.34162868224000000.1 |
| Degree 18 sibling: | 18.0.4641657809149437673472000000000.1 |
| Minimal sibling: | 12.0.34162868224000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.12.24.318 | $x^{12} + 10 x^{10} + 12 x^{9} + 110 x^{8} + 80 x^{7} + 752 x^{6} + 512 x^{5} + 1636 x^{4} + 1504 x^{3} + 1224 x^{2} + 1008 x - 648$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
|
\(5\)
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(19\)
| 19.18.12.1 | $x^{18} + 1938 x^{16} + 165 x^{15} + 1564986 x^{14} + 217074 x^{13} + 673956770 x^{12} + 124220751 x^{11} + 163260541175 x^{10} + 40698676942 x^{9} + 21101810097561 x^{8} + 7872565858164 x^{7} + 1139107203476720 x^{6} + 760256762812749 x^{5} + 441034593923007 x^{4} + 19443033036221259 x^{3} + 22753074450170540 x^{2} + 9283251966890258 x + 2743830934025437$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |