magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 9, 26, 75, 18, 28, 9, 9, -1, 3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 3*x^10 - x^9 + 9*x^8 + 9*x^7 + 28*x^6 + 18*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
gp: K = bnfinit(x^12 + 3*x^10 - x^9 + 9*x^8 + 9*x^7 + 28*x^6 + 18*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1, 1)
Normalized defining polynomial
\( x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1 \)
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: | \(84075626953125=3^{16}\cdot 5^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.47$ | 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: | $3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(45=3^{2}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{45}(1,·)$, $\chi_{45}(34,·)$, $\chi_{45}(4,·)$, $\chi_{45}(37,·)$, $\chi_{45}(7,·)$, $\chi_{45}(43,·)$, $\chi_{45}(13,·)$, $\chi_{45}(16,·)$, $\chi_{45}(19,·)$, $\chi_{45}(22,·)$, $\chi_{45}(28,·)$, $\chi_{45}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{701} a^{9} - \frac{9}{701} a^{8} + \frac{81}{701} a^{7} - \frac{28}{701} a^{6} + \frac{252}{701} a^{5} - \frac{75}{701} a^{4} - \frac{26}{701} a^{3} + \frac{234}{701} a^{2} - \frac{3}{701} a + \frac{27}{701}$, $\frac{1}{701} a^{10} + \frac{90}{701} a^{5} + \frac{243}{701}$, $\frac{1}{701} a^{11} + \frac{90}{701} a^{6} + \frac{243}{701} a$
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: | \( \frac{9}{701} a^{11} + \frac{109}{701} a^{6} - \frac{2019}{701} a \) (order $10$) | 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: | \( 201.000834787 \) | 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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{5})\), 6.6.820125.1 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }$ | R | R | ${\href{/LocalNumberField/7.12.0.1}{12} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.12.16.14 | $x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
| $5$ | 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
Artin representations
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.