Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 2*x^12 - x^11 + 3*x^10 - 4*x^9 + 3*x^8 - 4*x^7 + 2*x^6 - 4*x^5 + 2*x^4 + 5*x^2 + 2*x + 1)
gp: K = bnfinit(x^14 - x^13 + 2*x^12 - x^11 + 3*x^10 - 4*x^9 + 3*x^8 - 4*x^7 + 2*x^6 - 4*x^5 + 2*x^4 + 5*x^2 + 2*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 5, 0, 2, -4, 2, -4, 3, -4, 3, -1, 2, -1, 1]);
\( x^{14} - x^{13} + 2 x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 3 x^{8} - 4 x^{7} + 2 x^{6} - 4 x^{5} + 2 x^{4} + 5 x^{2} + 2 x + 1 \)
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: | \(-280155320935227\)\(\medspace = -\,3^{7}\cdot 71^{6}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $10.76$ | 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: | $3, 71$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7}$, $\frac{1}{119} a^{13} + \frac{2}{119} a^{12} - \frac{43}{119} a^{11} - \frac{45}{119} a^{10} + \frac{4}{119} a^{9} + \frac{59}{119} a^{8} + \frac{27}{119} a^{7} + \frac{9}{119} a^{6} + \frac{12}{119} a^{5} + \frac{32}{119} a^{4} - \frac{38}{119} a^{3} + \frac{22}{119} a^{2} + \frac{3}{119} a + \frac{11}{119}$
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: | \( -\frac{47}{119} a^{13} + \frac{6}{17} a^{12} - \frac{10}{17} a^{11} + \frac{1}{17} a^{10} - \frac{86}{119} a^{9} + \frac{134}{119} a^{8} - \frac{96}{119} a^{7} + \frac{138}{119} a^{6} - \frac{88}{119} a^{5} + \frac{128}{119} a^{4} - \frac{12}{17} a^{3} + \frac{20}{119} a^{2} - \frac{260}{119} a + \frac{10}{119} \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 64.9438034848 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 28 |
| The 10 conjugacy class representatives for $D_{14}$ |
| Character table for $D_{14}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | Deg 28 |
| Degree 14 sibling: | 14.2.19891027786401117.1 |
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 | ${\href{/LocalNumberField/5.14.0.1}{14} }$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.14.7.2 | $x^{14} + 243 x^{4} - 729 x^{2} + 2187$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
| $71$ | 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.2.1 | $x^{4} + 1491 x^{2} + 609961$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.213.2t1.a.a | $1$ | $ 3 \cdot 71 $ | \(\Q(\sqrt{213}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 2.71.7t2.a.c | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.71.7t2.a.a | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.639.14t3.a.a | $2$ | $ 3^{2} \cdot 71 $ | 14.0.280155320935227.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.71.7t2.a.b | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.639.14t3.a.b | $2$ | $ 3^{2} \cdot 71 $ | 14.0.280155320935227.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.639.14t3.a.c | $2$ | $ 3^{2} \cdot 71 $ | 14.0.280155320935227.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
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 *.