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Properties

Label 7.1.63425726272.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,2^{6}\cdot 997^{3}$
Root discriminant $34.93$
Ramified primes $2, 997$
Class number $13$
Class group $[13]$
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![96, 272, 44, 116, 1, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + 7*x^5 + x^4 + 116*x^3 + 44*x^2 + 272*x + 96)
 
gp: K = bnfinit(x^7 - x^6 + 7*x^5 + x^4 + 116*x^3 + 44*x^2 + 272*x + 96, 1)
 

Normalized defining polynomial

\( x^{7} - x^{6} + 7 x^{5} + x^{4} + 116 x^{3} + 44 x^{2} + 272 x + 96 \)

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:  \(-63425726272=-\,2^{6}\cdot 997^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.93$
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:  $2, 997$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{96} a^{5} + \frac{1}{12} a^{4} + \frac{3}{32} a^{3} + \frac{1}{48} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{192} a^{6} - \frac{1}{192} a^{5} + \frac{3}{64} a^{4} + \frac{17}{192} a^{3} - \frac{5}{96} a^{2} - \frac{1}{2} a + \frac{1}{4}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{13}$, which has order $13$

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:  \( \frac{3}{32} a^{6} - \frac{11}{96} a^{5} + \frac{65}{96} a^{4} - \frac{3}{32} a^{3} + \frac{505}{48} a^{2} + \frac{1}{3} a + \frac{49}{2} \),  \( \frac{13}{48} a^{6} - \frac{1}{2} a^{5} + \frac{101}{48} a^{4} - \frac{23}{24} a^{3} + \frac{185}{6} a^{2} - \frac{40}{3} a + 63 \),  \( \frac{1}{96} a^{6} + \frac{11}{96} a^{5} + \frac{3}{32} a^{4} + \frac{77}{96} a^{3} + \frac{79}{48} a^{2} + 10 a + \frac{7}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 605.69054128 \)
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: 14.0.16043017139485116019720192.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
997Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_997.2t1.1c1$1$ $ 2^{2} \cdot 997 $ $x^{2} + 997$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_997.7t2.1c1$2$ $ 2^{2} \cdot 997 $ $x^{7} - x^{6} + 7 x^{5} + x^{4} + 116 x^{3} + 44 x^{2} + 272 x + 96$ $D_{7}$ (as 7T2) $1$ $0$
* 2.2e2_997.7t2.1c2$2$ $ 2^{2} \cdot 997 $ $x^{7} - x^{6} + 7 x^{5} + x^{4} + 116 x^{3} + 44 x^{2} + 272 x + 96$ $D_{7}$ (as 7T2) $1$ $0$
* 2.2e2_997.7t2.1c3$2$ $ 2^{2} \cdot 997 $ $x^{7} - x^{6} + 7 x^{5} + x^{4} + 116 x^{3} + 44 x^{2} + 272 x + 96$ $D_{7}$ (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 *.