magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5637140613117, 0, 453563037837, 0, 10426736502, 0, 102726468, 0, 491985, 0, 1131, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 1131*x^10 + 491985*x^8 + 102726468*x^6 + 10426736502*x^4 + 453563037837*x^2 + 5637140613117)
gp: K = bnfinit(x^12 + 1131*x^10 + 491985*x^8 + 102726468*x^6 + 10426736502*x^4 + 453563037837*x^2 + 5637140613117, 1)
Normalized defining polynomial
\( x^{12} + 1131 x^{10} + 491985 x^{8} + 102726468 x^{6} + 10426736502 x^{4} + 453563037837 x^{2} + 5637140613117 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3183115072573672681403731968=2^{12}\cdot 3^{6}\cdot 13^{11}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $195.84$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 13, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4524=2^{2}\cdot 3\cdot 13\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4524}(1,·)$, $\chi_{4524}(3479,·)$, $\chi_{4524}(2785,·)$, $\chi_{4524}(2089,·)$, $\chi_{4524}(2087,·)$, $\chi_{4524}(1741,·)$, $\chi_{4524}(4175,·)$, $\chi_{4524}(4177,·)$, $\chi_{4524}(3827,·)$, $\chi_{4524}(695,·)$, $\chi_{4524}(3481,·)$, $\chi_{4524}(3131,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{87} a^{2}$, $\frac{1}{87} a^{3}$, $\frac{1}{7569} a^{4}$, $\frac{1}{7569} a^{5}$, $\frac{1}{658503} a^{6}$, $\frac{1}{658503} a^{7}$, $\frac{1}{57289761} a^{8}$, $\frac{1}{57289761} a^{9}$, $\frac{1}{4984209207} a^{10}$, $\frac{1}{4984209207} a^{11}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{162370}$, which has order $2597920$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $5$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{1}{57289761} a^{8} + \frac{1}{73167} a^{6} + \frac{3}{841} a^{4} + \frac{1}{3} a^{2} + 6 \), \( \frac{1}{57289761} a^{8} + \frac{8}{658503} a^{6} + \frac{20}{7569} a^{4} + \frac{16}{87} a^{2} + 2 \), \( \frac{1}{658503} a^{6} + \frac{2}{2523} a^{4} + \frac{3}{29} a^{2} + 2 \), \( \frac{1}{57289761} a^{8} + \frac{1}{73167} a^{6} + \frac{26}{7569} a^{4} + \frac{25}{87} a^{2} + 4 \), \( \frac{1}{658503} a^{6} + \frac{2}{2523} a^{4} + \frac{3}{29} a^{2} + 3 \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 120.78403136265631 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.266065488.4, \(\Q(\zeta_{13})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }$ | ${\href{/LocalNumberField/41.12.0.1}{12} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.12.25 | $x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $13$ | 13.12.11.4 | $x^{12} - 832$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |
| $29$ | 29.6.3.2 | $x^{6} - 841 x^{2} + 73167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 29.6.3.2 | $x^{6} - 841 x^{2} + 73167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |