# Properties

 Label 7.1.237751.1 Degree $7$ Signature $[1, 3]$ Discriminant $-\,23\cdot 10337$ Root discriminant $5.86$ Ramified primes $23, 10337$ Class number $1$ Class group Trivial Galois group $S_7$ (as 7T7)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 + x^5 - x^4 + x^3 + x^2 - 2*x + 1)

gp: K = bnfinit(x^7 - x^6 + x^5 - x^4 + x^3 + x^2 - 2*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 1, 1, -1, 1, -1, 1]);

## Normalizeddefining polynomial

$$x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $7$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[1, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-237751=-\,23\cdot 10337$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.86$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $23, 10337$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $3$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a$$,  $$a^{6} - a^{5} - a^{3} + a - 2$$,  $$2 a^{6} - a^{5} + a^{4} - 2 a^{3} + a^{2} + 3 a - 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.455031505787$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$S_7$ (as 7T7):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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: Deg 30 Degree 35 sibling: Deg 35 Degree 42 siblings: Deg 42, Deg 42, Deg 42, some 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 ${\href{/LocalNumberField/2.7.0.1}{7} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2} 23.5.0.1x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
10337Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.237751.2t1.a.a$1$ $23 \cdot 10337$ $x^{2} - x + 59438$ $C_2$ (as 2T1) $1$ $-1$
6.759646885244864506814313751.14t46.a.a$6$ $23^{5} \cdot 10337^{5}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
* 6.237751.7t7.a.a$6$ $23 \cdot 10337$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
14.3195136446302495076001.21t38.a.a$14$ $23^{4} \cdot 10337^{4}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $2$
14.577063390262224344757068055417048375398350897067690001.42t413.a.a$14$ $23^{10} \cdot 10337^{10}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-2$
14.2427175449366035662340297434782812166503404389751.30t565.a.a$14$ $23^{9} \cdot 10337^{9}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
14.759646885244864506814313751.30t565.a.a$14$ $23^{5} \cdot 10337^{5}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
15.759646885244864506814313751.42t412.a.a$15$ $23^{5} \cdot 10337^{5}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-3$
15.577063390262224344757068055417048375398350897067690001.42t411.a.a$15$ $23^{10} \cdot 10337^{10}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $3$
20.577063390262224344757068055417048375398350897067690001.70.a.a$20$ $23^{10} \cdot 10337^{10}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $0$
21.577063390262224344757068055417048375398350897067690001.84.a.a$21$ $23^{10} \cdot 10337^{10}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-3$
21.137197398098234100190337687243458668299333324128740365427751.42t418.a.a$21$ $23^{11} \cdot 10337^{11}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $3$
35.333002156380932238720846444022613283566064291985992138204507923338655046562464307036230104237143730235380001.126.a.a$35$ $23^{20} \cdot 10337^{20}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (as 7T7) $1$ $-1$
35.438364407001540399123028258559890113446389339483378620949520883861410051619503751.70.a.a$35$ $23^{15} \cdot 10337^{15}$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$ $S_7$ (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 *.