# Properties

 Label 8.4.37090522921.1 Degree $8$ Signature $[4, 2]$ Discriminant $29^{4}\cdot 229^{2}$ Root discriminant $20.95$ Ramified primes $29, 229$ Class number $1$ Class group Trivial Galois group $V_4^2:(S_3\times C_2)$ (as 8T41)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, -16, 8, 11, -4, 1, -3, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + x^6 - 4*x^5 + 11*x^4 + 8*x^3 - 16*x^2 + 2*x - 1)

gp: K = bnfinit(x^8 - 3*x^7 + x^6 - 4*x^5 + 11*x^4 + 8*x^3 - 16*x^2 + 2*x - 1, 1)

## Normalizeddefining polynomial

$$x^{8} - 3 x^{7} + x^{6} - 4 x^{5} + 11 x^{4} + 8 x^{3} - 16 x^{2} + 2 x - 1$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $8$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[4, 2]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$37090522921=29^{4}\cdot 229^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $20.95$ 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: $29, 229$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{55} a^{7} - \frac{4}{11} a^{6} + \frac{1}{5} a^{5} - \frac{26}{55} a^{4} + \frac{13}{55} a^{3} + \frac{7}{55} a^{2} - \frac{5}{11} a - \frac{13}{55}$

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: $5$ 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 magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$294.187101399$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_2^2:S_4:C_2$ (as 8T41):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 192 The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$ Character table for $V_4^2:(S_3\times C_2)$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Degree 8 sibling: data not computed Degree 12 siblings: data not computed Degree 16 siblings: data not computed Degree 24 siblings: data not computed Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2} 29.2.1.1x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
229Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.29_229.2t1.1c1$1$ $29 \cdot 229$ $x^{2} - x - 1660$ $C_2$ (as 2T1) $1$ $1$
* 1.29.2t1.1c1$1$ $29$ $x^{2} - x - 7$ $C_2$ (as 2T1) $1$ $1$
1.229.2t1.1c1$1$ $229$ $x^{2} - x - 57$ $C_2$ (as 2T1) $1$ $1$
2.29e2_229.6t3.2c1$2$ $29^{2} \cdot 229$ $x^{6} - 176 x^{4} - 65 x^{3} + 7744 x^{2} + 5720 x - 80296$ $D_{6}$ (as 6T3) $1$ $2$
2.229.3t2.1c1$2$ $229$ $x^{3} - 4 x - 1$ $S_3$ (as 3T2) $1$ $2$
3.229e2.6t8.1c1$3$ $229^{2}$ $x^{4} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.29e3_229e2.6t11.1c1$3$ $29^{3} \cdot 229^{2}$ $x^{6} - 3 x^{5} + 69 x^{4} - 133 x^{3} - 1846 x^{2} + 1912 x - 87749$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.29e3_229.6t11.1c1$3$ $29^{3} \cdot 229$ $x^{6} - 3 x^{5} + 69 x^{4} - 133 x^{3} - 1846 x^{2} + 1912 x - 87749$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.229.4t5.1c1$3$ $229$ $x^{4} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
* 6.29e3_229e2.8t41.1c1$6$ $29^{3} \cdot 229^{2}$ $x^{8} - 3 x^{7} + x^{6} - 4 x^{5} + 11 x^{4} + 8 x^{3} - 16 x^{2} + 2 x - 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.29e3_229e3.12t108.1c1$6$ $29^{3} \cdot 229^{3}$ $x^{8} - 3 x^{7} + x^{6} - 4 x^{5} + 11 x^{4} + 8 x^{3} - 16 x^{2} + 2 x - 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
6.29e3_229e3.8t41.1c1$6$ $29^{3} \cdot 229^{3}$ $x^{8} - 3 x^{7} + x^{6} - 4 x^{5} + 11 x^{4} + 8 x^{3} - 16 x^{2} + 2 x - 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$
6.29e3_229e4.12t108.1c1$6$ $29^{3} \cdot 229^{4}$ $x^{8} - 3 x^{7} + x^{6} - 4 x^{5} + 11 x^{4} + 8 x^{3} - 16 x^{2} + 2 x - 1$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$

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 *.