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Properties

Label 8.8.57178852641.1
Degree $8$
Signature $[8, 0]$
Discriminant $3^{4}\cdot 163^{4}$
Root discriminant $22.11$
Ramified primes $3, 163$
Class number $1$
Class group Trivial
Galois group $\SL(2,3)$ (as 8T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 9, -38, -73, 48, 24, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 13*x^6 + 24*x^5 + 48*x^4 - 73*x^3 - 38*x^2 + 9*x + 3)
 
gp: K = bnfinit(x^8 - 2*x^7 - 13*x^6 + 24*x^5 + 48*x^4 - 73*x^3 - 38*x^2 + 9*x + 3, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 73 x^{3} - 38 x^{2} + 9 x + 3 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57178852641=3^{4}\cdot 163^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.11$
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:  $3, 163$
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 \(a\))

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

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:  $7$
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:  \( 751.205508883 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\SL(2,3)$ (as 8T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 7 conjugacy class representatives for $\SL(2,3)$
Character table for $\SL(2,3)$

Intermediate fields

4.4.26569.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$163$163.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
163.6.4.1$x^{6} + 5216 x^{3} + 35363339$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.163.3t1.1c1$1$ $ 163 $ $x^{3} - x^{2} - 54 x + 169$ $C_3$ (as 3T1) $0$ $1$
1.163.3t1.1c2$1$ $ 163 $ $x^{3} - x^{2} - 54 x + 169$ $C_3$ (as 3T1) $0$ $1$
2.3e2_163e2.24t7.1c1$2$ $ 3^{2} \cdot 163^{2}$ $x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 73 x^{3} - 38 x^{2} + 9 x + 3$ $\SL(2,3)$ (as 8T12) $-1$ $2$
* 2.3e2_163.8t12.1c1$2$ $ 3^{2} \cdot 163 $ $x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 73 x^{3} - 38 x^{2} + 9 x + 3$ $\SL(2,3)$ (as 8T12) $0$ $2$
* 2.3e2_163.8t12.1c2$2$ $ 3^{2} \cdot 163 $ $x^{8} - 2 x^{7} - 13 x^{6} + 24 x^{5} + 48 x^{4} - 73 x^{3} - 38 x^{2} + 9 x + 3$ $\SL(2,3)$ (as 8T12) $0$ $2$
* 3.163e2.4t4.1c1$3$ $ 163^{2}$ $x^{4} - x^{3} - 7 x^{2} + 2 x + 9$ $A_4$ (as 4T4) $1$ $3$

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 *.