Properties

Label 7.1.211831.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,19\cdot 11149$
Root discriminant $5.77$
Ramified primes $19, 11149$
Class number $1$
Class group Trivial
Galois Group $S_7$ (as 7T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 1, 0, -1, 2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + 2*x^5 - x^4 + x^2 - 2*x + 1)
gp: K = bnfinit(x^7 - x^6 + 2*x^5 - x^4 + x^2 - 2*x + 1, 1)

Normalized defining polynomial

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

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:  \(-211831=-\,19\cdot 11149\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $5.77$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $19, 11149$
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 Abelian group, 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:  None
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.418943361216 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_7$ (as 7T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 5040
The 15 conjugacy class representatives for $S_7$
Character table for $S_7$

Intermediate fields

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

Sibling fields

Degree 14 sibling: Deg 14
Degree 21 sibling: Deg 21
Degree 30 sibling: data not computed
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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11149Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.19_11149.2t1.1c1$1$ $ 19 \cdot 11149 $ $x^{2} - x + 52958$ $C_2$ (as 2T1) $1$ $-1$
6.19e5_11149e5.14t46.1c1$6$ $ 19^{5} \cdot 11149^{5}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
6.19_11149.7t7.1c1$6$ $ 19 \cdot 11149 $ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
14.19e4_11149e4.21t38.1c1$14$ $ 19^{4} \cdot 11149^{4}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $2$
14.19e10_11149e10.42t413.1c1$14$ $ 19^{10} \cdot 11149^{10}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-2$
14.19e9_11149e9.30t565.1c1$14$ $ 19^{9} \cdot 11149^{9}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
14.19e5_11149e5.30t565.1c1$14$ $ 19^{5} \cdot 11149^{5}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
15.19e5_11149e5.42t412.1c1$15$ $ 19^{5} \cdot 11149^{5}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-3$
15.19e10_11149e10.42t411.1c1$15$ $ 19^{10} \cdot 11149^{10}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $3$
20.19e10_11149e10.70.1c1$20$ $ 19^{10} \cdot 11149^{10}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
21.19e10_11149e10.84.1c1$21$ $ 19^{10} \cdot 11149^{10}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-3$
21.19e11_11149e11.42t418.1c1$21$ $ 19^{11} \cdot 11149^{11}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $3$
35.19e20_11149e20.126.1c1$35$ $ 19^{20} \cdot 11149^{20}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-1$
35.19e15_11149e15.70.1c1$35$ $ 19^{15} \cdot 11149^{15}$ $x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $1$

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 *.