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Properties

Label 8.4.2310148096.2
Degree $8$
Signature $[4, 2]$
Discriminant $2^{12}\cdot 751^{2}$
Root discriminant $14.81$
Ramified primes $2, 751$
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![-7, 24, -10, -30, 15, 14, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 7*x^6 + 14*x^5 + 15*x^4 - 30*x^3 - 10*x^2 + 24*x - 7)
 
gp: K = bnfinit(x^8 - 2*x^7 - 7*x^6 + 14*x^5 + 15*x^4 - 30*x^3 - 10*x^2 + 24*x - 7, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 30 x^{3} - 10 x^{2} + 24 x - 7 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2310148096=2^{12}\cdot 751^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.81$
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:  $2, 751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

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:  $5$
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:  \( a^{7} - 2 a^{6} - 8 a^{5} + 14 a^{4} + 22 a^{3} - 28 a^{2} - 25 a + 17 \),  \( a^{7} - 2 a^{6} - 8 a^{5} + 14 a^{4} + 22 a^{3} - 29 a^{2} - 24 a + 19 \),  \( a^{7} - 2 a^{6} - 8 a^{5} + 14 a^{4} + 22 a^{3} - 28 a^{2} - 24 a + 18 \),  \( a^{7} - 7 a^{5} + 15 a^{3} - 10 a + 4 \),  \( a^{3} - a^{2} - 2 a + 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 84.8125764638 \)
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

\(\Q(\sqrt{2}) \)

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
751Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e3_751.2t1.2c1$1$ $ 2^{3} \cdot 751 $ $x^{2} + 1502$ $C_2$ (as 2T1) $1$ $-1$
1.751.2t1.1c1$1$ $ 751 $ $x^{2} - x + 188$ $C_2$ (as 2T1) $1$ $-1$
2.2e6_751.6t3.2c1$2$ $ 2^{6} \cdot 751 $ $x^{6} + 68 x^{4} + 1156 x^{2} + 6008$ $D_{6}$ (as 6T3) $1$ $0$
2.751.3t2.1c1$2$ $ 751 $ $x^{3} - x^{2} + 6 x - 1$ $S_3$ (as 3T2) $1$ $0$
3.751e2.6t8.1c1$3$ $ 751^{2}$ $x^{4} - 2 x^{3} + x^{2} - x - 1$ $S_4$ (as 4T5) $1$ $-1$
3.2e9_751.6t11.2c1$3$ $ 2^{9} \cdot 751 $ $x^{6} + 6 x^{4} + 760 x^{2} + 6008$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.2e9_751e2.6t11.2c1$3$ $ 2^{9} \cdot 751^{2}$ $x^{6} + 6 x^{4} + 760 x^{2} + 6008$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.751.4t5.1c1$3$ $ 751 $ $x^{4} - 2 x^{3} + x^{2} - x - 1$ $S_4$ (as 4T5) $1$ $1$
* 6.2e9_751e2.8t41.1c1$6$ $ 2^{9} \cdot 751^{2}$ $x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 30 x^{3} - 10 x^{2} + 24 x - 7$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $2$
6.2e9_751e3.12t108.1c1$6$ $ 2^{9} \cdot 751^{3}$ $x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 30 x^{3} - 10 x^{2} + 24 x - 7$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
6.2e9_751e3.8t41.1c1$6$ $ 2^{9} \cdot 751^{3}$ $x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 30 x^{3} - 10 x^{2} + 24 x - 7$ $V_4^2:(S_3\times C_2)$ (as 8T41) $1$ $0$
6.2e9_751e4.12t108.1c1$6$ $ 2^{9} \cdot 751^{4}$ $x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 30 x^{3} - 10 x^{2} + 24 x - 7$ $V_4^2:(S_3\times C_2)$ (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 *.