Properties

Label 8.2.82104483840.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{16}\cdot 3\cdot 5\cdot 17^{4}$
Root discriminant $23.14$
Ramified primes $2, 3, 5, 17$
Class number $2$
Class group $[2]$
Galois group $S_4\wr C_2$ (as 8T47)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4*x^6 - 4*x^5 - 18*x^4 - 8*x^3 - 40*x^2 + 44*x - 15)
 
gp: K = bnfinit(x^8 + 4*x^6 - 4*x^5 - 18*x^4 - 8*x^3 - 40*x^2 + 44*x - 15, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 44, -40, -8, -18, -4, 4, 0, 1]);
 

Normalized defining polynomial

\( x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-82104483840=-\,2^{16}\cdot 3\cdot 5\cdot 17^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.14$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{1740} a^{7} + \frac{169}{1740} a^{6} - \frac{1}{12} a^{5} - \frac{149}{1740} a^{4} - \frac{839}{1740} a^{3} - \frac{859}{1740} a^{2} + \frac{79}{1740} a - \frac{35}{116}$

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:  $4$
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:  \( 519.058800592 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

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

\(\Q(\sqrt{34}) \)

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 ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.16.58$x^{8} + 2 x^{6} + 4$$8$$1$$16$$S_4\times C_2$$[4/3, 4/3, 3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$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$1$ $1$ $x$ $C_1$ $1$ $1$
1.15.2t1.a.a$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
1.2040.2t1.b.a$1$ $ 2^{3} \cdot 3 \cdot 5 \cdot 17 $ $x^{2} + 510$ $C_2$ (as 2T1) $1$ $-1$
* 1.136.2t1.a.a$1$ $ 2^{3} \cdot 17 $ $x^{2} - 34$ $C_2$ (as 2T1) $1$ $1$
2.2040.4t3.l.a$2$ $ 2^{3} \cdot 3 \cdot 5 \cdot 17 $ $x^{4} + x^{2} + 34$ $D_{4}$ (as 4T3) $1$ $0$
4.122400.6t13.b.a$4$ $ 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 17 $ $x^{6} - 3 x^{5} + 4 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.249696000.12t34.h.a$4$ $ 2^{8} \cdot 3^{3} \cdot 5^{3} \cdot 17^{2}$ $x^{6} - 3 x^{5} + 4 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.565977600.12t34.b.a$4$ $ 2^{9} \cdot 3^{2} \cdot 5^{2} \cdot 17^{3}$ $x^{6} - 3 x^{5} + 4 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.1109760.6t13.b.a$4$ $ 2^{8} \cdot 3 \cdot 5 \cdot 17^{2}$ $x^{6} - 3 x^{5} + 4 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.2037519360000.12t201.a.a$6$ $ 2^{13} \cdot 3^{4} \cdot 5^{4} \cdot 17^{3}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.30562790400000.12t202.a.a$6$ $ 2^{13} \cdot 3^{5} \cdot 5^{5} \cdot 17^{3}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.603709440.8t47.a.a$6$ $ 2^{13} \cdot 3 \cdot 5 \cdot 17^{3}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.9055641600.12t200.a.a$6$ $ 2^{13} \cdot 3^{2} \cdot 5^{2} \cdot 17^{3}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.543338496000.16t1294.a.a$9$ $ 2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.1833767424000000.18t272.a.a$9$ $ 2^{15} \cdot 3^{6} \cdot 5^{6} \cdot 17^{3}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.1153190317326336000000.18t273.a.a$9$ $ 2^{22} \cdot 3^{6} \cdot 5^{6} \cdot 17^{6}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.341686019948544000.18t274.a.a$9$ $ 2^{22} \cdot 3^{3} \cdot 5^{3} \cdot 17^{6}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.276765676158320640000000.36t1763.a.a$12$ $ 2^{26} \cdot 3^{7} \cdot 5^{7} \cdot 17^{6}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.1230069671814758400000.24t2821.a.a$12$ $ 2^{26} \cdot 3^{5} \cdot 5^{5} \cdot 17^{6}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.626572692617854143430656000000000.36t1758.a.a$18$ $ 2^{37} \cdot 3^{9} \cdot 5^{9} \cdot 17^{9}$ $x^{8} + 4 x^{6} - 4 x^{5} - 18 x^{4} - 8 x^{3} - 40 x^{2} + 44 x - 15$ $S_4\wr C_2$ (as 8T47) $1$ $0$

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 *.