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Properties

Label 8.0.433347489.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{8}\cdot 257^{2}$
Root discriminant $12.01$
Ramified primes $3, 257$
Class number $2$
Class group $[2]$
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![21, -51, 58, -45, 27, -13, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 6*x^6 - 13*x^5 + 27*x^4 - 45*x^3 + 58*x^2 - 51*x + 21)
 
gp: K = bnfinit(x^8 - 3*x^7 + 6*x^6 - 13*x^5 + 27*x^4 - 45*x^3 + 58*x^2 - 51*x + 21, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 6 x^{6} - 13 x^{5} + 27 x^{4} - 45 x^{3} + 58 x^{2} - 51 x + 21 \)

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:  \(433347489=3^{8}\cdot 257^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.01$
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:  $3, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{45} a^{7} - \frac{4}{45} a^{6} - \frac{1}{9} a^{5} + \frac{7}{45} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{2}{5} a + \frac{7}{15}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{4}{15} a^{7} + \frac{2}{5} a^{6} - a^{5} + \frac{32}{15} a^{4} - 4 a^{3} + 6 a^{2} - \frac{34}{5} a + \frac{22}{5} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a - 1 \),  \( \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{5}{9} a^{5} - \frac{13}{9} a^{4} + \frac{19}{9} a^{3} - \frac{28}{9} a^{2} + 3 a - \frac{1}{3} \),  \( \frac{4}{45} a^{7} - \frac{1}{45} a^{6} + \frac{2}{9} a^{5} - \frac{17}{45} a^{4} + \frac{10}{9} a^{3} - \frac{13}{9} a^{2} + \frac{8}{5} a - \frac{2}{15} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26.1752814701 \)
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$$[\ ]_{2}$
3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
257Data not computed