# Properties

 Label 8.0.565732361546641.1 Degree $8$ Signature $[0, 4]$ Discriminant $4877^{4}$ Root discriminant $69.84$ Ramified prime $4877$ Class number $6$ (GRH) Class group $[6]$ (GRH) Galois group $\PSL(2,7)$ (as 8T37)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1700, -1708, 1347, -569, 259, -71, 25, -3, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 25*x^6 - 71*x^5 + 259*x^4 - 569*x^3 + 1347*x^2 - 1708*x + 1700)

gp: K = bnfinit(x^8 - 3*x^7 + 25*x^6 - 71*x^5 + 259*x^4 - 569*x^3 + 1347*x^2 - 1708*x + 1700, 1)

## Normalizeddefining polynomial

$$x^{8} - 3 x^{7} + 25 x^{6} - 71 x^{5} + 259 x^{4} - 569 x^{3} + 1347 x^{2} - 1708 x + 1700$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $8$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 4]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$565732361546641=4877^{4}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $69.84$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $4877$ magma: PrimeDivisors(Discriminant(Integers(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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{307532} a^{7} - \frac{7989}{307532} a^{6} + \frac{141055}{307532} a^{5} + \frac{24415}{307532} a^{4} - \frac{2643}{307532} a^{3} - \frac{113279}{307532} a^{2} - \frac{111703}{307532} a - \frac{45941}{153766}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{6}$, which has order $6$ (assuming GRH)

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: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$14757.7931566$$ (assuming GRH) magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$\PSL(2,7)$ (as 8T37):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A non-solvable group of order 168 The 6 conjugacy class representatives for $\PSL(2,7)$ Character table for $\PSL(2,7)$

## Intermediate fields

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

## Sibling fields

 Degree 7 siblings: data not computed Degree 14 siblings: data not computed Degree 21 sibling: data not computed Degree 24 sibling: data not computed Degree 28 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
4877Data not computed