Properties

Label 7.1.364871.1
Degree $7$
Signature $[1, 3]$
Discriminant $-\,13^{2}\cdot 17\cdot 127$
Root discriminant $6.23$
Ramified primes $13, 17, 127$
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, 0, 0, 2, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + 2*x^4 + 2*x + 1)
gp: K = bnfinit(x^7 - x^6 + 2*x^4 + 2*x + 1, 1)

Normalized defining polynomial

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

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:  \(-364871=-\,13^{2}\cdot 17\cdot 127\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $6.23$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $13, 17, 127$
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:  \( a^{6} - a^{5} + 2 a^{3} - a^{2} + a + 2 \),  \( a \),  \( a^{6} - 2 a^{5} + a^{4} + 2 a^{3} - 2 a^{2} + 2 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.631311355925 \)
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
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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R R ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.17_127.2t1.1c1$1$ $ 17 \cdot 127 $ $x^{2} - x + 540$ $C_2$ (as 2T1) $1$ $-1$
6.13e2_17e5_127e5.14t46.1c1$6$ $ 13^{2} \cdot 17^{5} \cdot 127^{5}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
* 6.13e2_17_127.7t7.1c1$6$ $ 13^{2} \cdot 17 \cdot 127 $ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
14.13e6_17e4_127e4.21t38.1c1$14$ $ 13^{6} \cdot 17^{4} \cdot 127^{4}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $2$
14.13e6_17e10_127e10.42t413.1c1$14$ $ 13^{6} \cdot 17^{10} \cdot 127^{10}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $-2$
14.13e6_17e9_127e9.30t565.1c1$14$ $ 13^{6} \cdot 17^{9} \cdot 127^{9}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
14.13e6_17e5_127e5.30t565.1c1$14$ $ 13^{6} \cdot 17^{5} \cdot 127^{5}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
15.13e8_17e5_127e5.42t412.1c1$15$ $ 13^{8} \cdot 17^{5} \cdot 127^{5}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $-3$
15.13e8_17e10_127e10.42t411.1c1$15$ $ 13^{8} \cdot 17^{10} \cdot 127^{10}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $3$
20.13e12_17e10_127e10.70.1c1$20$ $ 13^{12} \cdot 17^{10} \cdot 127^{10}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
21.13e10_17e10_127e10.84.1c1$21$ $ 13^{10} \cdot 17^{10} \cdot 127^{10}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $-3$
21.13e10_17e11_127e11.42t418.1c1$21$ $ 13^{10} \cdot 17^{11} \cdot 127^{11}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $3$
35.13e18_17e20_127e20.126.1c1$35$ $ 13^{18} \cdot 17^{20} \cdot 127^{20}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (as 7T7) $1$ $-1$
35.13e18_17e15_127e15.70.1c1$35$ $ 13^{18} \cdot 17^{15} \cdot 127^{15}$ $x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ $S_7$ (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 *.