magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30, -28, -42, -21, -14, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 14*x^4 - 21*x^3 - 42*x^2 - 28*x + 30)
gp: K = bnfinit(x^7 - 14*x^4 - 21*x^3 - 42*x^2 - 28*x + 30, 1)
Normalized defining polynomial
\( x^{7} - 14 x^{4} - 21 x^{3} - 42 x^{2} - 28 x + 30 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3320525376=2^{6}\cdot 3^{2}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.92$ | 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: | $2, 3, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{5} - \frac{4}{27} a^{4} - \frac{4}{27} a^{3} + \frac{1}{27} a^{2} + \frac{1}{27} a - \frac{4}{9}$, $\frac{1}{27} a^{6} - \frac{2}{27} a^{4} + \frac{1}{9} a^{3} - \frac{13}{27} a^{2} - \frac{8}{27} a + \frac{2}{9}$
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: | $4$ | 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: | 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: | \( 712.578223565 \) | 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: | 8.0.119538913536.1 |
| 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: | 7.3.3320525376.3 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.7.0.1}{7} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.7.0.1}{7} }$ | ${\href{/LocalNumberField/43.7.0.1}{7} }$ | ${\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.7.8.3 | $x^{7} + 28 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ |
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_3e2_7e4.42t37.2c1 | $3$ | $ 2^{4} \cdot 3^{2} \cdot 7^{4}$ | $x^{7} - 14 x^{4} - 21 x^{3} - 42 x^{2} - 28 x + 30$ | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
| 3.2e4_3e2_7e4.42t37.2c2 | $3$ | $ 2^{4} \cdot 3^{2} \cdot 7^{4}$ | $x^{7} - 14 x^{4} - 21 x^{3} - 42 x^{2} - 28 x + 30$ | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
| * | 6.2e6_3e2_7e8.7t5.2c1 | $6$ | $ 2^{6} \cdot 3^{2} \cdot 7^{8}$ | $x^{7} - 14 x^{4} - 21 x^{3} - 42 x^{2} - 28 x + 30$ | $\GL(3,2)$ (as 7T5) | $1$ | $2$ |
| 7.2e8_3e4_7e8.8t37.2c1 | $7$ | $ 2^{8} \cdot 3^{4} \cdot 7^{8}$ | $x^{7} - 14 x^{4} - 21 x^{3} - 42 x^{2} - 28 x + 30$ | $\GL(3,2)$ (as 7T5) | $1$ | $-1$ | |
| 8.2e10_3e4_7e10.21t14.2c1 | $8$ | $ 2^{10} \cdot 3^{4} \cdot 7^{10}$ | $x^{7} - 14 x^{4} - 21 x^{3} - 42 x^{2} - 28 x + 30$ | $\GL(3,2)$ (as 7T5) | $1$ | $0$ |
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 *.