Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - x^12 + 2*x^10 + x^9 - x^8 - x^7 - x^6 + x^5 + 2*x^4 - x^2 - x + 1)
gp: K = bnfinit(x^14 - x^13 - x^12 + 2*x^10 + x^9 - x^8 - x^7 - x^6 + x^5 + 2*x^4 - x^2 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -1, 0, 2, 1, -1, -1, -1, 1, 2, 0, -1, -1, 1]);
\( x^{14} - x^{13} - x^{12} + 2 x^{10} + x^{9} - x^{8} - x^{7} - x^{6} + x^{5} + 2 x^{4} - x^{2} - x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-172575877556547\)\(\medspace = -\,3^{7}\cdot 280909^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $10.40$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $3, 280909$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $2$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( a^{13} - a^{11} - 2 a^{10} + a^{9} + 3 a^{8} + a^{7} - 2 a^{5} - a^{4} + a^{3} + 2 a^{2} + a - 1 \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | \( a^{13} - a^{11} - a^{10} + 2 a^{8} + 2 a^{7} - a^{5} - a^{4} + a^{3} + a^{2} + a \), \( a^{12} - a^{11} - a^{10} + a^{8} + 2 a^{7} - a^{5} - 2 a^{4} + 2 a^{2} \), \( a^{13} - a^{12} + a^{8} + a^{6} - a^{5} + a^{3} + a \), \( a^{11} - a^{10} - a^{9} + a^{8} + a^{6} - a^{3} + a \), \( a^{12} - 2 a^{10} + a^{8} + 2 a^{7} + a^{6} - a^{5} - a^{4} - a^{3} + 2 a^{2} + a \), \( a^{12} - a^{10} - a^{9} + 2 a^{7} + a^{6} + a^{5} - a^{4} - a^{3} + a + 1 \) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 55.4764206676 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$S_7\times C_2$ (as 14T49):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A non-solvable group of order 10080 |
| The 30 conjugacy class representatives for $S_7\times C_2$ |
| Character table for $S_7\times C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 7.5.7584543.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 14 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 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 | ${\href{/LocalNumberField/2.14.0.1}{14} }$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 280909 | Data not computed | ||||||