Properties

Label 7.1.357911.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,71^{3}$
Root discriminant $6.21$
Ramified prime $71$
Class number $1$
Class group Trivial
Galois Group $D_{7}$ (as 7T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -1, -1, 1, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - x^5 + x^4 - x^3 - x^2 + 2*x + 1)
gp: K = bnfinit(x^7 - x^6 - x^5 + x^4 - x^3 - x^2 + 2*x + 1, 1)

Normalized defining polynomial

\(x^{7} \) \(\mathstrut -\mathstrut x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut +\mathstrut x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut -\mathstrut x^{2} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $7$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 3]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-357911=-\,71^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $6.21$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $71$
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:  $3$
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:  \( a^{5} - a^{3} + a^{2} - a - 2 \),  \( a^{6} - a^{5} + a^{3} - 2 a^{2} - a + 1 \),  \( a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.604934372036 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_7$ (as 7T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

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

Sibling fields

Galois closure: 14.0.9095120158391.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.71.2t1.1c1$1$ $ 71 $ $x^{2} - x + 18$ $C_2$ (as 2T1) $1$ $-1$
* 2.71.7t2.1c1$2$ $ 71 $ $x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.1c2$2$ $ 71 $ $x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.1c3$2$ $ 71 $ $x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1$ $D_{7}$ (as 7T2) $1$ $0$

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 *.