magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -12, 30, -44, 57, -64, 60, -48, 34, -20, 10, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 20*x^9 + 34*x^8 - 48*x^7 + 60*x^6 - 64*x^5 + 57*x^4 - 44*x^3 + 30*x^2 - 12*x + 2)
gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 20*x^9 + 34*x^8 - 48*x^7 + 60*x^6 - 64*x^5 + 57*x^4 - 44*x^3 + 30*x^2 - 12*x + 2, 1)
Normalized defining polynomial
\( x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 34 x^{8} - 48 x^{7} + 60 x^{6} - 64 x^{5} + 57 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2 \)
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: | \(10312216477696=2^{32}\cdot 7^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.15$ | 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: | $2, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{41} a^{11} + \frac{20}{41} a^{10} - \frac{2}{41} a^{9} + \frac{14}{41} a^{8} + \frac{1}{41} a^{7} + \frac{17}{41} a^{6} + \frac{17}{41} a^{5} + \frac{16}{41} a^{4} - \frac{10}{41} a^{3} + \frac{3}{41} a^{2} + \frac{20}{41} a + \frac{17}{41}$
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: | \( \frac{26}{41} a^{11} - \frac{95}{41} a^{10} + \frac{235}{41} a^{9} - \frac{456}{41} a^{8} + \frac{764}{41} a^{7} - \frac{1034}{41} a^{6} + \frac{1262}{41} a^{5} - \frac{1265}{41} a^{4} + \frac{1052}{41} a^{3} - \frac{742}{41} a^{2} + \frac{479}{41} a - \frac{91}{41} \) (order $8$) | 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: | \( 347.441365493 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_3:S_3.C_2$ (as 12T40):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 72 |
| The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$ |
| Character table for $C_2\times C_3:S_3.C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\), 6.2.802816.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
| 2.8.24.10 | $x^{8} + 16$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ | |
| $7$ | 7.6.4.2 | $x^{6} - 7 x^{3} + 147$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |