magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -8, -14, 31, 0, -12, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 12*x^5 + 31*x^3 - 14*x^2 - 8*x + 4)
gp: K = bnfinit(x^7 - 12*x^5 + 31*x^3 - 14*x^2 - 8*x + 4, 1)
Normalized defining polynomial
\( x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[7, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(18424261696=2^{6}\cdot 19^{4}\cdot 47^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.27$ | 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: | $2, 19, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$
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: | $6$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{7}{2} a^{6} + 2 a^{5} - 41 a^{4} - 23 a^{3} + \frac{193}{2} a^{2} + 3 a - 27 \), \( 5 a^{6} + 3 a^{5} - 58 a^{4} - 35 a^{3} + 132 a^{2} + 11 a - 31 \), \( a^{2} + a - 1 \), \( a^{2} + a - 3 \), \( \frac{1}{2} a^{6} + a^{5} - 7 a^{4} - 9 a^{3} + \frac{41}{2} a^{2} + 3 a - 5 \), \( \frac{7}{2} a^{6} + 2 a^{5} - 41 a^{4} - 24 a^{3} + \frac{191}{2} a^{2} + 9 a - 23 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2113.84591487 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$\PSL(2,7)$ (as 7T5):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 168 |
| The 6 conjugacy class representatives for $\GL(3,2)$ |
| Character table for $\GL(3,2)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 8 sibling: | 8.8.58769636260854016.1 |
| Degree 14 siblings: | Deg 14, Deg 14 |
| Degree 21 sibling: | Deg 21 |
| Degree 24 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently sibling: | 7.7.18424261696.2 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.4.4.5 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.3.1 | $x^{4} + 76$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $47$ | $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.4.2.2 | $x^{4} - 47 x^{2} + 28717$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 3.2e4_19e2_47e2.42t37.1c1 | $3$ | $ 2^{4} \cdot 19^{2} \cdot 47^{2}$ | $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ | $\GL(3,2)$ (as 7T5) | $0$ | $3$ | |
| 3.2e4_19e2_47e2.42t37.1c2 | $3$ | $ 2^{4} \cdot 19^{2} \cdot 47^{2}$ | $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ | $\GL(3,2)$ (as 7T5) | $0$ | $3$ | |
| * | 6.2e6_19e4_47e2.7t5.1c1 | $6$ | $ 2^{6} \cdot 19^{4} \cdot 47^{2}$ | $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ | $\GL(3,2)$ (as 7T5) | $1$ | $6$ |
| 7.2e8_19e6_47e4.8t37.1c1 | $7$ | $ 2^{8} \cdot 19^{6} \cdot 47^{4}$ | $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ | $\GL(3,2)$ (as 7T5) | $1$ | $7$ | |
| 8.2e10_19e6_47e4.21t14.1c1 | $8$ | $ 2^{10} \cdot 19^{6} \cdot 47^{4}$ | $x^{7} - 12 x^{5} + 31 x^{3} - 14 x^{2} - 8 x + 4$ | $\GL(3,2)$ (as 7T5) | $1$ | $8$ |
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 *.