# Properties

 Label 8.4.65289632040.1 Degree $8$ Signature $[4, 2]$ Discriminant $65289632040$ Root discriminant $22.48$ Ramified primes $2, 3, 5, 67$ Class number $1$ Class group trivial Galois group $S_4\wr C_2$ (as 8T47)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 3*x^6 + 11*x^5 - 5*x^4 - 12*x^3 - 34*x^2 - 7*x + 1)

gp: K = bnfinit(x^8 - 2*x^7 - 3*x^6 + 11*x^5 - 5*x^4 - 12*x^3 - 34*x^2 - 7*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, -34, -12, -5, 11, -3, -2, 1]);

$$x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $8$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$65289632040$$$$\medspace = 2^{3}\cdot 3^{4}\cdot 5\cdot 67^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $22.48$ 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: $2, 3, 5, 67$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4295} a^{7} - \frac{2001}{4295} a^{6} + \frac{1351}{4295} a^{5} + \frac{917}{4295} a^{4} + \frac{877}{4295} a^{3} - \frac{155}{859} a^{2} - \frac{1304}{4295} a - \frac{376}{4295}$

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: $5$ 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: 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$952.155611166$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{2}\cdot 952.155611166 \cdot 1}{2\sqrt{65289632040}}\approx 1.17688911015$

## Galois group

$S_4\wr C_2$ (as 8T47):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 1152 The 20 conjugacy class representatives for $S_4\wr C_2$ Character table for $S_4\wr C_2$

## Intermediate fields

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

## Sibling fields

 Degree 12 siblings: data not computed Degree 16 siblings: data not computed Degree 18 siblings: data not computed Degree 24 siblings: data not computed Degree 32 siblings: data not computed Degree 36 siblings: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3] 2.2.0.1x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4} 33.2.1.1x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3} 6767.2.1.2x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.40.2t1.a.a$1$ $2^{3} \cdot 5$ $x^{2} - 10$ $C_2$ (as 2T1) $1$ $1$
1.8040.2t1.b.a$1$ $2^{3} \cdot 3 \cdot 5 \cdot 67$ $x^{2} - 2010$ $C_2$ (as 2T1) $1$ $1$
* 1.201.2t1.a.a$1$ $3 \cdot 67$ $x^{2} - x - 50$ $C_2$ (as 2T1) $1$ $1$
2.8040.4t3.b.a$2$ $2^{3} \cdot 3 \cdot 5 \cdot 67$ $x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 15$ $D_{4}$ (as 4T3) $1$ $-2$
4.12992961600.12t34.g.a$4$ $2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 67^{3}$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2585664000.12t34.c.a$4$ $2^{9} \cdot 3^{2} \cdot 5^{3} \cdot 67^{2}$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.321600.6t13.e.a$4$ $2^{6} \cdot 3 \cdot 5^{2} \cdot 67$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.1616040.6t13.c.a$4$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 67^{2}$ $x^{6} - 6 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
6.207...000.12t201.a.a$6$ $2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.831...000.12t202.a.a$6$ $2^{15} \cdot 3^{3} \cdot 5^{5} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $2$
* 6.324824040.8t47.a.a$6$ $2^{3} \cdot 3^{3} \cdot 5 \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.12992961600.12t200.a.a$6$ $2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.519718464000.16t1294.a.a$9$ $2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.332...000.18t272.a.a$9$ $2^{18} \cdot 3^{3} \cdot 5^{6} \cdot 67^{3}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.270...000.18t273.a.a$9$ $2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.422...000.18t274.a.a$9$ $2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
12.108...000.36t1763.a.a$12$ $2^{21} \cdot 3^{6} \cdot 5^{7} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $0$
12.675...000.24t2821.a.a$12$ $2^{15} \cdot 3^{6} \cdot 5^{5} \cdot 67^{6}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $0$
18.140...000.36t1758.a.a$18$ $2^{27} \cdot 3^{9} \cdot 5^{9} \cdot 67^{9}$ $x^{8} - 2 x^{7} - 3 x^{6} + 11 x^{5} - 5 x^{4} - 12 x^{3} - 34 x^{2} - 7 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$

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 *.