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Properties

Label 7.1.642735647.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,863^{3}$
Root discriminant $18.13$
Ramified prime $863$
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, 16, -24, 10, 6, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 4*x^5 + 6*x^4 + 10*x^3 - 24*x^2 + 16*x + 1)
 
gp: K = bnfinit(x^7 - x^6 - 4*x^5 + 6*x^4 + 10*x^3 - 24*x^2 + 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} - 4 x^{5} + 6 x^{4} + 10 x^{3} - 24 x^{2} + 16 x + 1 \)

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

Invariants

Degree:  $7$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-642735647=-\,863^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.13$
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:  $863$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{35} a^{6} - \frac{13}{35} a^{5} + \frac{12}{35} a^{4} + \frac{2}{35} a^{3} - \frac{2}{5} a^{2} + \frac{4}{35} a + \frac{3}{35}$

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:  \( \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + a^{2} - \frac{10}{7} a + \frac{10}{7} \),  \( \frac{9}{35} a^{6} - \frac{12}{35} a^{5} - \frac{32}{35} a^{4} + \frac{53}{35} a^{3} + \frac{12}{5} a^{2} - \frac{209}{35} a + \frac{132}{35} \),  \( \frac{11}{35} a^{6} - \frac{3}{35} a^{5} - \frac{43}{35} a^{4} + \frac{22}{35} a^{3} + \frac{13}{5} a^{2} - \frac{166}{35} a + \frac{138}{35} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 40.4319760493 \)
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: 14.0.356513163590850929567.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }$

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
863Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.863.2t1.1c1$1$ $ 863 $ $x^{2} - x + 216$ $C_2$ (as 2T1) $1$ $-1$
* 2.863.7t2.1c1$2$ $ 863 $ $x^{7} - x^{6} - 4 x^{5} + 6 x^{4} + 10 x^{3} - 24 x^{2} + 16 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.863.7t2.1c2$2$ $ 863 $ $x^{7} - x^{6} - 4 x^{5} + 6 x^{4} + 10 x^{3} - 24 x^{2} + 16 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.863.7t2.1c3$2$ $ 863 $ $x^{7} - x^{6} - 4 x^{5} + 6 x^{4} + 10 x^{3} - 24 x^{2} + 16 x + 1$ $D_{7}$ (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 *.