Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)
gp: K = bnfinit(x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1]);
\( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(1706902865139206151939937338729\)\(\medspace = 3^{12}\cdot 13^{22}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $18.18$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $3, 13$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $24$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(39=3\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{39}(1,·)$, $\chi_{39}(2,·)$, $\chi_{39}(4,·)$, $\chi_{39}(5,·)$, $\chi_{39}(7,·)$, $\chi_{39}(8,·)$, $\chi_{39}(10,·)$, $\chi_{39}(11,·)$, $\chi_{39}(14,·)$, $\chi_{39}(16,·)$, $\chi_{39}(17,·)$, $\chi_{39}(19,·)$, $\chi_{39}(20,·)$, $\chi_{39}(22,·)$, $\chi_{39}(23,·)$, $\chi_{39}(25,·)$, $\chi_{39}(28,·)$, $\chi_{39}(29,·)$, $\chi_{39}(31,·)$, $\chi_{39}(32,·)$, $\chi_{39}(34,·)$, $\chi_{39}(35,·)$, $\chi_{39}(37,·)$, $\chi_{39}(38,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -a \) (order $78$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | \( a^{3} + 1 \), \( a^{6} + 1 \), \( a^{17} - a^{15} + a^{4} - 1 \), \( a^{6} + a^{3} + 1 \), \( a^{21} + a^{3} \), \( a - 1 \), \( a^{2} - 1 \), \( a^{4} - 1 \), \( a^{5} - 1 \), \( a^{7} - 1 \), \( a^{10} - 1 \) (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 2851634.018949717 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2\times C_{12}$ (as 24T2):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| An abelian group of order 24 |
| The 24 conjugacy class representatives for $C_2\times C_{12}$ |
| Character table for $C_2\times C_{12}$ is not computed |
Intermediate fields
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 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 | $x^{12} - 832$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |
| 13.12.11.4 | $x^{12} - 832$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |