magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 4, 2, -1, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 3*x^5 - x^4 + 2*x^3 + 4*x^2 + 4*x + 1)
gp: K = bnfinit(x^7 - x^6 - 3*x^5 - x^4 + 2*x^3 + 4*x^2 + 4*x + 1, 1)
Normalized defining polynomial
\( x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1 \)
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: | \(15864289=7^{2}\cdot 569^{2}\) | magma: Discriminant(K);
sage: K.disc()
gp: K.disc
| |
Root discriminant: | $10.68$ | magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
Ramified primes: | $7, 569$ | magma: PrimeDivisors(Discriminant(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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$
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: | \( a \), \( a^{6} - 2 a^{5} - a^{4} + 2 a^{2} + 2 a + 1 \), \( a^{6} - a^{5} - 3 a^{4} + a^{2} + 3 a + 2 \), \( 2 a^{6} - 3 a^{5} - 4 a^{4} + 3 a^{2} + 6 a + 4 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
Regulator: | \( 8.28712673096 \) | 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: | 8.0.251675665475521.3 |
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: | 7.3.15864289.1 |
Frobenius cycle types
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/LocalNumberField/2.7.0.1}{7} }$ | ${\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
$7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
569 | Data not computed |
Artin representations
Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
3.7e2_569e2.42t37.2c1 | $3$ | $ 7^{2} \cdot 569^{2}$ | $x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1$ | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
3.7e2_569e2.42t37.2c2 | $3$ | $ 7^{2} \cdot 569^{2}$ | $x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1$ | $\GL(3,2)$ (as 7T5) | $0$ | $-1$ | |
* | 6.7e2_569e2.7t5.2c1 | $6$ | $ 7^{2} \cdot 569^{2}$ | $x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1$ | $\GL(3,2)$ (as 7T5) | $1$ | $2$ |
7.7e4_569e4.8t37.2c1 | $7$ | $ 7^{4} \cdot 569^{4}$ | $x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1$ | $\GL(3,2)$ (as 7T5) | $1$ | $-1$ | |
8.7e4_569e4.21t14.2c1 | $8$ | $ 7^{4} \cdot 569^{4}$ | $x^{7} - x^{6} - 3 x^{5} - x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 1$ | $\GL(3,2)$ (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 *.