magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 216, 0, 2250, 0, 9393, 0, 21087, 0, 28743, 0, 25298, 0, 14823, 0, 5832, 0, 1521, 0, 252, 0, 24, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 24*x^22 + 252*x^20 + 1521*x^18 + 5832*x^16 + 14823*x^14 + 25298*x^12 + 28743*x^10 + 21087*x^8 + 9393*x^6 + 2250*x^4 + 216*x^2 + 1)
gp: K = bnfinit(x^24 + 24*x^22 + 252*x^20 + 1521*x^18 + 5832*x^16 + 14823*x^14 + 25298*x^12 + 28743*x^10 + 21087*x^8 + 9393*x^6 + 2250*x^4 + 216*x^2 + 1, 1)
Normalized defining polynomial
\( x^{24} + 24 x^{22} + 252 x^{20} + 1521 x^{18} + 5832 x^{16} + 14823 x^{14} + 25298 x^{12} + 28743 x^{10} + 21087 x^{8} + 9393 x^{6} + 2250 x^{4} + 216 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 12]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9606056659007943744000000000000000000=2^{24}\cdot 3^{36}\cdot 5^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.75$ | 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: | $2, 3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(180=2^{2}\cdot 3^{2}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(113,·)$, $\chi_{180}(137,·)$, $\chi_{180}(139,·)$, $\chi_{180}(109,·)$, $\chi_{180}(77,·)$, $\chi_{180}(143,·)$, $\chi_{180}(17,·)$, $\chi_{180}(19,·)$, $\chi_{180}(23,·)$, $\chi_{180}(79,·)$, $\chi_{180}(31,·)$, $\chi_{180}(91,·)$, $\chi_{180}(167,·)$, $\chi_{180}(169,·)$, $\chi_{180}(107,·)$, $\chi_{180}(173,·)$, $\chi_{180}(47,·)$, $\chi_{180}(49,·)$, $\chi_{180}(83,·)$, $\chi_{180}(53,·)$, $\chi_{180}(121,·)$, $\chi_{180}(151,·)$, $\chi_{180}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{5}\times C_{30}$, which has order $150$ (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: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( a^{15} + 15 a^{13} + 90 a^{11} + 275 a^{9} + 450 a^{7} + 378 a^{5} + 140 a^{3} + 15 a \) (order $4$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 4249664.47225837 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{12}$ (as 24T2):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| 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 | R | R | R | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 | Data not computed | ||||||
| $3$ | 3.12.18.74 | $x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$ | $6$ | $2$ | $18$ | $C_{12}$ | $[2]_{2}^{2}$ |
| 3.12.18.74 | $x^{12} - 6 x^{11} - 3 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{7} + 12 x^{6} - 9 x^{3} - 9$ | $6$ | $2$ | $18$ | $C_{12}$ | $[2]_{2}^{2}$ | |
| $5$ | 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |