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Properties

Label 8.0.79227801.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{4}\cdot 23^{2}\cdot 43^{2}$
Root discriminant $9.71$
Ramified primes $3, 23, 43$
Class number $1$
Class group Trivial
Galois group $V_4^2:(S_3\times C_2)$ (as 8T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -9, 10, -6, 4, -4, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 4*x^6 - 4*x^5 + 4*x^4 - 6*x^3 + 10*x^2 - 9*x + 3)
 
gp: K = bnfinit(x^8 - 2*x^7 + 4*x^6 - 4*x^5 + 4*x^4 - 6*x^3 + 10*x^2 - 9*x + 3, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 4 x^{6} - 4 x^{5} + 4 x^{4} - 6 x^{3} + 10 x^{2} - 9 x + 3 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(79227801=3^{4}\cdot 23^{2}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.71$
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:  $3, 23, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} + \frac{4}{13} a^{5} + \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{5}{13} a^{2} - \frac{6}{13} a + \frac{5}{13}$

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:  \( -\frac{12}{13} a^{7} + a^{6} - \frac{35}{13} a^{5} + \frac{17}{13} a^{4} - \frac{27}{13} a^{3} + \frac{44}{13} a^{2} - \frac{71}{13} a + \frac{44}{13} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a - 1 \),  \( \frac{11}{13} a^{7} - a^{6} + \frac{31}{13} a^{5} - \frac{21}{13} a^{4} + \frac{28}{13} a^{3} - \frac{49}{13} a^{2} + \frac{77}{13} a - \frac{49}{13} \),  \( \frac{1}{13} a^{7} + \frac{4}{13} a^{5} + \frac{4}{13} a^{4} + \frac{12}{13} a^{3} + \frac{5}{13} a^{2} + \frac{7}{13} a - \frac{8}{13} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15.6673169956 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4:C_2$ (as 8T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$
Character table for $V_4^2:(S_3\times C_2)$

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Sibling fields

Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ R ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_23.2t1.1c1$1$ $ 3 \cdot 23 $ $x^{2} - x - 17$ $C_2$ (as 2T1) $1$ $1$
1.23.2t1.1c1$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
2.23.3t2.1c1$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
2.3e2_23.6t3.1c1$2$ $ 3^{2} \cdot 23 $ $x^{6} - x^{5} + x^{4} - 2 x^{3} + x^{2} + 1$ $D_{6}$ (as 6T3) $1$ $0$
3.23e2_43e2.6t8.1c1$3$ $ 23^{2} \cdot 43^{2}$ $x^{4} - x^{3} - x^{2} + 6 x - 7$ $S_4$ (as 4T5) $1$ $-1$
3.3e3_23_43e2.6t11.1c1$3$ $ 3^{3} \cdot 23 \cdot 43^{2}$ $x^{6} - 2 x^{5} - 5 x^{4} + 80 x^{3} + 73 x^{2} - 1533 x - 17123$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.23_43e2.4t5.1c1$3$ $ 23 \cdot 43^{2}$ $x^{4} - x^{3} - x^{2} + 6 x - 7$ $S_4$ (as 4T5) $1$ $1$
3.3e3_23e2_43e2.6t11.1c1$3$ $ 3^{3} \cdot 23^{2} \cdot 43^{2}$ $x^{6} - 2 x^{5} - 5 x^{4} + 80 x^{3} + 73 x^{2} - 1533 x - 17123$ $S_4\times C_2$ (as 6T11) $1$ $1$
6.3e3_23e3_43e4.8t41.2c1$6$ $ 3^{3} \cdot 23^{3} \cdot 43^{4}$ $x^{8} - 2 x^{7} + 4 x^{6} - 4 x^{5} + 4 x^{4} - 6 x^{3} + 10 x^{2} - 9 x + 3$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
* 6.3e3_23e2_43e2.8t41.2c1$6$ $ 3^{3} \cdot 23^{2} \cdot 43^{2}$ $x^{8} - 2 x^{7} + 4 x^{6} - 4 x^{5} + 4 x^{4} - 6 x^{3} + 10 x^{2} - 9 x + 3$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
6.3e3_23e4_43e2.12t111.1c1$6$ $ 3^{3} \cdot 23^{4} \cdot 43^{2}$ $x^{8} - 2 x^{7} + 4 x^{6} - 4 x^{5} + 4 x^{4} - 6 x^{3} + 10 x^{2} - 9 x + 3$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
6.3e3_23e3_43e4.12t108.2c1$6$ $ 3^{3} \cdot 23^{3} \cdot 43^{4}$ $x^{8} - 2 x^{7} + 4 x^{6} - 4 x^{5} + 4 x^{4} - 6 x^{3} + 10 x^{2} - 9 x + 3$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $-2$

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 *.