Properties

Label 7.3.16711744.1
Degree $7$
Signature $[3, 2]$
Discriminant $2^{6}\cdot 7^{2}\cdot 73^{2}$
Root discriminant $10.76$
Ramified primes $2, 7, 73$
Class number $1$
Class group Trivial
Galois Group $A_7$ (as 7T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 2, -2, -2, 2, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 2*x^6 + 2*x^4 - 2*x^3 - 2*x^2 + 2*x + 2)
gp: K = bnfinit(x^7 - 2*x^6 + 2*x^4 - 2*x^3 - 2*x^2 + 2*x + 2, 1)

Normalized defining polynomial

\(x^{7} \) \(\mathstrut -\mathstrut 2 x^{6} \) \(\mathstrut +\mathstrut 2 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut -\mathstrut 2 x^{2} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 2 \)

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:  \(16711744=2^{6}\cdot 7^{2}\cdot 73^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.76$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 73$
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:  \( a^{6} - 2 a^{5} + a^{4} + a^{3} - 2 a^{2} - a + 1 \),  \( a^{6} - 2 a^{5} + a^{4} - a^{2} - a + 1 \),  \( a^{2} - 1 \),  \( a - 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 13.3010937939 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$A_7$ (as 7T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 2520
The 9 conjugacy class representatives for $A_7$
Character table for $A_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 15 siblings: Deg 15, Deg 15
Degree 21 sibling: Deg 21
Degree 35 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 R ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
73Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 6.2e6_7e2_73e2.7t6.1c1$6$ $ 2^{6} \cdot 7^{2} \cdot 73^{2}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $1$ $2$
10.2e9_7e6_73e6.70.1c1$10$ $ 2^{9} \cdot 7^{6} \cdot 73^{6}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $0$ $-2$
10.2e9_7e6_73e6.70.1c2$10$ $ 2^{9} \cdot 7^{6} \cdot 73^{6}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $0$ $-2$
14.2e12_7e6_73e10.15t47.1c1$14$ $ 2^{12} \cdot 7^{6} \cdot 73^{10}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $1$ $2$
14.2e12_7e6_73e8.21t33.1c1$14$ $ 2^{12} \cdot 7^{6} \cdot 73^{8}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $1$ $2$
15.2e12_7e8_73e8.42t294.1c1$15$ $ 2^{12} \cdot 7^{8} \cdot 73^{8}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $1$ $-1$
21.2e18_7e10_73e16.42t299.1c1$21$ $ 2^{18} \cdot 7^{10} \cdot 73^{16}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $1$ $1$
35.2e30_7e18_73e24.70.1c1$35$ $ 2^{30} \cdot 7^{18} \cdot 73^{24}$ $x^{7} - 2 x^{6} + 2 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 2$ $A_7$ (as 7T6) $1$ $-1$

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 *.