magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -10, 12, -8, 5, -12, 12, 0, 3, -6, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^9 + 3*x^8 + 12*x^6 - 12*x^5 + 5*x^4 - 8*x^3 + 12*x^2 - 10*x + 5)
gp: K = bnfinit(x^12 - 6*x^9 + 3*x^8 + 12*x^6 - 12*x^5 + 5*x^4 - 8*x^3 + 12*x^2 - 10*x + 5, 1)
Normalized defining polynomial
\( x^{12} - 6 x^{9} + 3 x^{8} + 12 x^{6} - 12 x^{5} + 5 x^{4} - 8 x^{3} + 12 x^{2} - 10 x + 5 \)
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: | \(7867596800000=2^{20}\cdot 5^{5}\cdot 7^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $11.88$ | 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: | $2, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{355} a^{11} - \frac{113}{355} a^{10} - \frac{11}{355} a^{9} + \frac{172}{355} a^{8} + \frac{92}{355} a^{7} - \frac{101}{355} a^{6} + \frac{13}{71} a^{5} + \frac{98}{355} a^{4} - \frac{64}{355} a^{3} + \frac{124}{355} a^{2} - \frac{31}{71} a + \frac{22}{71}$
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: | \( -\frac{20}{71} a^{11} - \frac{12}{71} a^{10} + \frac{7}{71} a^{9} + \frac{110}{71} a^{8} + \frac{6}{71} a^{7} - \frac{39}{71} a^{6} - \frac{164}{71} a^{5} + \frac{99}{71} a^{4} + \frac{2}{71} a^{3} + \frac{5}{71} a^{2} - \frac{95}{71} a + \frac{72}{71} \) (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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 114.898797006 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\times D_4$ (as 12T28):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 48 |
| The 15 conjugacy class representatives for $S_3\times D_4$ |
| Character table for $S_3\times D_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.1.140.1, 4.0.320.1, 6.0.313600.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.12.0.1}{12} }$ | R | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.20.68 | $x^{12} + 2 x^{9} + 2$ | $12$ | $1$ | $20$ | $(C_6\times C_2):C_2$ | $[2, 2]_{3}^{2}$ |
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |