↑

Properties

Label 8.0.966698534834176.6
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 17^{4}\cdot 41^{4}$
Root discriminant $74.67$
Ramified primes $2, 17, 41$
Class number $24$ (GRH)
Class group $[24]$ (GRH)
Galois group $V_4^2:S_3$ (as 8T34)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![693, -944, 300, -32, 122, -16, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4*x^6 - 16*x^5 + 122*x^4 - 32*x^3 + 300*x^2 - 944*x + 693)
 
gp: K = bnfinit(x^8 + 4*x^6 - 16*x^5 + 122*x^4 - 32*x^3 + 300*x^2 - 944*x + 693, 1)
 

Normalized defining polynomial

\( x^{8} + 4 x^{6} - 16 x^{5} + 122 x^{4} - 32 x^{3} + 300 x^{2} - 944 x + 693 \)

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:  \(966698534834176=2^{12}\cdot 17^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.67$
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:  $2, 17, 41$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{176} a^{6} + \frac{5}{88} a^{5} + \frac{3}{176} a^{4} + \frac{1}{11} a^{3} + \frac{23}{176} a^{2} - \frac{5}{88} a - \frac{1}{16}$, $\frac{1}{352} a^{7} - \frac{1}{352} a^{6} - \frac{19}{352} a^{5} + \frac{27}{352} a^{4} + \frac{23}{352} a^{3} - \frac{87}{352} a^{2} - \frac{15}{32} a - \frac{9}{32}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{24}$, which has order $24$ (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:  \( \frac{1}{32} a^{7} + \frac{3}{32} a^{6} + \frac{9}{32} a^{5} + \frac{3}{32} a^{4} + \frac{103}{32} a^{3} + \frac{293}{32} a^{2} + \frac{751}{32} a + \frac{533}{32} \),  \( \frac{5}{2} a^{4} + 5 a^{2} - 20 a + \frac{31}{2} \),  \( \frac{15927}{352} a^{7} + \frac{86621}{352} a^{6} + \frac{288983}{352} a^{5} + \frac{239109}{352} a^{4} - \frac{80271}{352} a^{3} - \frac{1133237}{352} a^{2} + \frac{47713}{352} a + \frac{108889}{32} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4952.93775628 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4$ (as 8T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 10 conjugacy class representatives for $V_4^2:S_3$
Character table for $V_4^2:S_3$

Intermediate fields

\(\Q(\sqrt{697}) \)

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 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.17_41.2t1.1c1$1$ $ 17 \cdot 41 $ $x^{2} - x - 174$ $C_2$ (as 2T1) $1$ $1$
2.17_41.3t2.1c1$2$ $ 17 \cdot 41 $ $x^{3} - 7 x - 5$ $S_3$ (as 3T2) $1$ $2$
3.2e6_17e2_41e2.6t8.2c1$3$ $ 2^{6} \cdot 17^{2} \cdot 41^{2}$ $x^{4} + 6 x^{2} - 8 x + 25$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_17_41.4t5.1c1$3$ $ 2^{6} \cdot 17 \cdot 41 $ $x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 3$ $S_4$ (as 4T5) $1$ $3$
3.17e2_41e2.6t8.2c1$3$ $ 17^{2} \cdot 41^{2}$ $x^{4} - x^{3} + 2 x^{2} - x + 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_17_41.4t5.2c1$3$ $ 2^{6} \cdot 17 \cdot 41 $ $x^{4} + 6 x^{2} - 8 x + 25$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_17e2_41e2.6t8.1c1$3$ $ 2^{6} \cdot 17^{2} \cdot 41^{2}$ $x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 3$ $S_4$ (as 4T5) $1$ $3$
3.17_41.4t5.1c1$3$ $ 17 \cdot 41 $ $x^{4} - x^{3} + 2 x^{2} - x + 2$ $S_4$ (as 4T5) $1$ $-1$
* 6.2e12_17e3_41e3.8t34.1c1$6$ $ 2^{12} \cdot 17^{3} \cdot 41^{3}$ $x^{8} + 4 x^{6} - 16 x^{5} + 122 x^{4} - 32 x^{3} + 300 x^{2} - 944 x + 693$ $V_4^2:S_3$ (as 8T34) $1$ $-2$

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 *.