magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 32, -64, -528, 784, 3142, -4088, -9189, 11034, 15333, -17391, -15810, 17187, 10506, -11067, -4573, 4709, 1294, -1312, -229, 230, 23, -23, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*x + 1)
gp: K = bnfinit(x^24 - x^23 - 23*x^22 + 23*x^21 + 230*x^20 - 229*x^19 - 1312*x^18 + 1294*x^17 + 4709*x^16 - 4573*x^15 - 11067*x^14 + 10506*x^13 + 17187*x^12 - 15810*x^11 - 17391*x^10 + 15333*x^9 + 11034*x^8 - 9189*x^7 - 4088*x^6 + 3142*x^5 + 784*x^4 - 528*x^3 - 64*x^2 + 32*x + 1, 1)
Normalized defining polynomial
\( x^{24} - x^{23} - 23 x^{22} + 23 x^{21} + 230 x^{20} - 229 x^{19} - 1312 x^{18} + 1294 x^{17} + 4709 x^{16} - 4573 x^{15} - 11067 x^{14} + 10506 x^{13} + 17187 x^{12} - 15810 x^{11} - 17391 x^{10} + 15333 x^{9} + 11034 x^{8} - 9189 x^{7} - 4088 x^{6} + 3142 x^{5} + 784 x^{4} - 528 x^{3} - 64 x^{2} + 32 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[24, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(161761786626698377317203521728515625=3^{12}\cdot 5^{18}\cdot 7^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.31$ | 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: | $3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(105=3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(4,·)$, $\chi_{105}(8,·)$, $\chi_{105}(73,·)$, $\chi_{105}(13,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(82,·)$, $\chi_{105}(23,·)$, $\chi_{105}(89,·)$, $\chi_{105}(26,·)$, $\chi_{105}(92,·)$, $\chi_{105}(32,·)$, $\chi_{105}(97,·)$, $\chi_{105}(101,·)$, $\chi_{105}(103,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(46,·)$, $\chi_{105}(52,·)$, $\chi_{105}(53,·)$, $\chi_{105}(59,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (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: | $23$ | 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: | 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: | \( 5459008653.485176 \) (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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | R | R | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $5$ | 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.1 | $x^{12} - 10 x^{8} - 375 x^{4} - 2000$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
| 7 | Data not computed | ||||||