# Properties

 Label 8.0.9597924961.1 Degree $8$ Signature $[0, 4]$ Discriminant $313^{4}$ Root discriminant $17.69$ Ramified prime $313$ Class number $2$ Class group $[2]$ Galois group $C_2^3:(C_7: C_3)$ (as 8T36)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + x^6 + 3*x^5 + 5*x^4 - 5*x^3 + 7*x + 9)

gp: K = bnfinit(x^8 - x^7 + x^6 + 3*x^5 + 5*x^4 - 5*x^3 + 7*x + 9, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 7, 0, -5, 5, 3, 1, -1, 1]);

## Normalizeddefining polynomial

$$x^{8} - x^{7} + x^{6} + 3 x^{5} + 5 x^{4} - 5 x^{3} + 7 x + 9$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$9597924961=313^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $17.69$ 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: $313$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

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: $$\frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + a + \frac{1}{2}$$,  $$\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{7}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$$,  $$\frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{4}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{2}{3} a + \frac{7}{2}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$59.1447316068$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$F_8:C_3$ (as 8T36):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 168 The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$ Character table for $C_2^3:(C_7: C_3)$

## Intermediate fields

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

## Sibling fields

 Degree 14 sibling: data not computed Degree 24 sibling: data not computed Degree 28 sibling: data not computed Degree 42 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
313Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.313.3t1.a.a$1$ $313$ $x^{3} - x^{2} - 104 x - 371$ $C_3$ (as 3T1) $0$ $1$
1.313.3t1.a.b$1$ $313$ $x^{3} - x^{2} - 104 x - 371$ $C_3$ (as 3T1) $0$ $1$
3.97969.7t3.a.a$3$ $313^{2}$ $x^{7} - x^{6} - 15 x^{5} + 20 x^{4} + 33 x^{3} - 22 x^{2} - 32 x - 8$ $C_7:C_3$ (as 7T3) $0$ $3$
3.97969.7t3.a.b$3$ $313^{2}$ $x^{7} - x^{6} - 15 x^{5} + 20 x^{4} + 33 x^{3} - 22 x^{2} - 32 x - 8$ $C_7:C_3$ (as 7T3) $0$ $3$
* 7.9597924961.8t36.b.a$7$ $313^{4}$ $x^{8} - x^{7} + x^{6} + 3 x^{5} + 5 x^{4} - 5 x^{3} + 7 x + 9$ $C_2^3:(C_7: C_3)$ (as 8T36) $1$ $-1$
7.3004150512793.24t283.b.a$7$ $313^{5}$ $x^{8} - x^{7} + x^{6} + 3 x^{5} + 5 x^{4} - 5 x^{3} + 7 x + 9$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$
7.3004150512793.24t283.b.b$7$ $313^{5}$ $x^{8} - x^{7} + x^{6} + 3 x^{5} + 5 x^{4} - 5 x^{3} + 7 x + 9$ $C_2^3:(C_7: C_3)$ (as 8T36) $0$ $-1$

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 *.