magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, 54, 0, 145, 0, 130, 0, 50, 0, 10, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 10*x^10 + 50*x^8 + 130*x^6 + 145*x^4 + 54*x^2 + 25)
gp: K = bnfinit(x^12 + 10*x^10 + 50*x^8 + 130*x^6 + 145*x^4 + 54*x^2 + 25, 1)
Normalized defining polynomial
\( x^{12} + 10 x^{10} + 50 x^{8} + 130 x^{6} + 145 x^{4} + 54 x^{2} + 25 \)
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: | \(2560000000000000000=2^{24}\cdot 5^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.20$ | 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: | $2, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1537} a^{10} + \frac{176}{1537} a^{8} + \frac{63}{1537} a^{6} - \frac{171}{1537} a^{4} - \frac{575}{1537} a^{2} - \frac{102}{1537}$, $\frac{1}{7685} a^{11} + \frac{650}{1537} a^{9} + \frac{320}{1537} a^{7} - \frac{649}{1537} a^{5} - \frac{115}{1537} a^{3} - \frac{3176}{7685} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$
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: | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 46625.1969434 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times A_5$ (as 12T76):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 120 |
| The 10 conjugacy class representatives for $C_2\times A_5$ |
| Character table for $C_2\times A_5$ |
Intermediate fields
| 6.2.25000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.24.280 | $x^{12} - 8 x^{11} + 4 x^{10} + 8 x^{9} - 6 x^{8} - 8 x^{6} - 8 x^{5} - 8 x^{4} - 8 x^{2} + 16 x - 8$ | $4$ | $3$ | $24$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.5.8.6 | $x^{5} + 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $D_{5}$ | $[2]^{2}$ | |
| 5.5.8.6 | $x^{5} + 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $D_{5}$ | $[2]^{2}$ |