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Properties

Label 8.0.2576952773521.1
Degree $8$
Signature $[0, 4]$
Discriminant $7^{4}\cdot 181^{4}$
Root discriminant $35.59$
Ramified primes $7, 181$
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![281, -440, 642, 13, -4, 12, -8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 8*x^6 + 12*x^5 - 4*x^4 + 13*x^3 + 642*x^2 - 440*x + 281)
 
gp: K = bnfinit(x^8 - x^7 - 8*x^6 + 12*x^5 - 4*x^4 + 13*x^3 + 642*x^2 - 440*x + 281, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 8 x^{6} + 12 x^{5} - 4 x^{4} + 13 x^{3} + 642 x^{2} - 440 x + 281 \)

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:  \(2576952773521=7^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.59$
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:  $7, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14} a^{5} - \frac{1}{7} a^{4} + \frac{3}{14} a^{3} + \frac{3}{7} a^{2} + \frac{1}{14} a - \frac{1}{14}$, $\frac{1}{28} a^{6} - \frac{1}{28} a^{5} + \frac{1}{28} a^{4} - \frac{5}{28} a^{3} - \frac{1}{4} a^{2} - \frac{1}{28}$, $\frac{1}{1537592} a^{7} - \frac{257}{27457} a^{6} - \frac{2543}{192199} a^{5} + \frac{14585}{54914} a^{4} - \frac{39995}{192199} a^{3} - \frac{543667}{1537592} a^{2} + \frac{424687}{1537592} a + \frac{476799}{1537592}$

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:  $3$
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:  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:  \( 474.774181023 \)
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

4.4.1605289.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7_181.3t1.2c1$1$ $ 7 \cdot 181 $ $x^{3} - x^{2} - 422 x + 3144$ $C_3$ (as 3T1) $0$ $1$
1.7_181.3t1.2c2$1$ $ 7 \cdot 181 $ $x^{3} - x^{2} - 422 x + 3144$ $C_3$ (as 3T1) $0$ $1$
2.7e2_181e2.24t7.1c1$2$ $ 7^{2} \cdot 181^{2}$ $x^{8} - x^{7} - 8 x^{6} + 12 x^{5} - 4 x^{4} + 13 x^{3} + 642 x^{2} - 440 x + 281$ $\SL(2,3)$ (as 8T12) $-1$ $-2$
* 2.7_181.8t12.1c1$2$ $ 7 \cdot 181 $ $x^{8} - x^{7} - 8 x^{6} + 12 x^{5} - 4 x^{4} + 13 x^{3} + 642 x^{2} - 440 x + 281$ $\SL(2,3)$ (as 8T12) $0$ $-2$
* 2.7_181.8t12.1c2$2$ $ 7 \cdot 181 $ $x^{8} - x^{7} - 8 x^{6} + 12 x^{5} - 4 x^{4} + 13 x^{3} + 642 x^{2} - 440 x + 281$ $\SL(2,3)$ (as 8T12) $0$ $-2$
* 3.7e2_181e2.4t4.2c1$3$ $ 7^{2} \cdot 181^{2}$ $x^{4} - x^{3} - 31 x^{2} - 42 x + 36$ $A_4$ (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 *.