# Properties

 Label 7.1.207911.1 Degree $7$ Signature $[1, 3]$ Discriminant $-\,11\cdot 41\cdot 461$ Root discriminant $5.75$ Ramified primes $11, 41, 461$ Class number $1$ Class group Trivial Galois Group $S_7$ (as 7T7)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 1, -1, -1, -1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^5 - x^4 - x^3 + x^2 + x + 1)
gp: K = bnfinit(x^7 - x^5 - x^4 - x^3 + x^2 + x + 1, 1)

## Normalizeddefining polynomial

$$x^{7}$$ $$\mathstrut -\mathstrut x^{5}$$ $$\mathstrut -\mathstrut x^{4}$$ $$\mathstrut -\mathstrut x^{3}$$ $$\mathstrut +\mathstrut x^{2}$$ $$\mathstrut +\mathstrut x$$ $$\mathstrut +\mathstrut 1$$

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: $$-207911=-\,11\cdot 41\cdot 461$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $5.75$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $11, 41, 461$ 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: $$a$$,  $$a^{6} - a^{4} - a^{2}$$,  $$a^{5} - a^{3}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$0.413665504587$$ 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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2} 11.5.0.1x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
41Data not computed
461Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.11_41_461.2t1.1c1$1$ $11 \cdot 41 \cdot 461$ $x^{2} - x + 51978$ $C_2$ (as 2T1) $1$ $-1$
6.11e5_41e5_461e5.14t46.1c1$6$ $11^{5} \cdot 41^{5} \cdot 461^{5}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $0$
* 6.11_41_461.7t7.1c1$6$ $11 \cdot 41 \cdot 461$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $0$
14.11e4_41e4_461e4.21t38.1c1$14$ $11^{4} \cdot 41^{4} \cdot 461^{4}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $2$
14.11e10_41e10_461e10.42t413.1c1$14$ $11^{10} \cdot 41^{10} \cdot 461^{10}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $-2$
14.11e9_41e9_461e9.30t565.1c1$14$ $11^{9} \cdot 41^{9} \cdot 461^{9}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $0$
14.11e5_41e5_461e5.30t565.1c1$14$ $11^{5} \cdot 41^{5} \cdot 461^{5}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $0$
15.11e5_41e5_461e5.42t412.1c1$15$ $11^{5} \cdot 41^{5} \cdot 461^{5}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $-3$
15.11e10_41e10_461e10.42t411.1c1$15$ $11^{10} \cdot 41^{10} \cdot 461^{10}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $3$
20.11e10_41e10_461e10.70.1c1$20$ $11^{10} \cdot 41^{10} \cdot 461^{10}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $0$
21.11e10_41e10_461e10.84.1c1$21$ $11^{10} \cdot 41^{10} \cdot 461^{10}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $-3$
21.11e11_41e11_461e11.42t418.1c1$21$ $11^{11} \cdot 41^{11} \cdot 461^{11}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $3$
35.11e20_41e20_461e20.126.1c1$35$ $11^{20} \cdot 41^{20} \cdot 461^{20}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (as 7T7) $1$ $-1$
35.11e15_41e15_461e15.70.1c1$35$ $11^{15} \cdot 41^{15} \cdot 461^{15}$ $x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ $S_7$ (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 *.