Properties

Label 8.0.16019205489.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{4}\cdot 7^{6}\cdot 41^{2}$
Root discriminant $18.86$
Ramified primes $3, 7, 41$
Class number $2$
Class group $[2]$
Galois group $C_2^4:C_6$ (as 8T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 4, 7, 0, -7, 7, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 7*x^5 - 7*x^4 + 7*x^2 + 4*x + 16)
 
gp: K = bnfinit(x^8 - 3*x^7 + 7*x^5 - 7*x^4 + 7*x^2 + 4*x + 16, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 7 x^{5} - 7 x^{4} + 7 x^{2} + 4 x + 16 \)

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:  \(16019205489=3^{4}\cdot 7^{6}\cdot 41^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.86$
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:  $3, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{84} a^{7} + \frac{5}{84} a^{6} - \frac{5}{21} a^{5} + \frac{1}{28} a^{4} - \frac{19}{84} a^{3} - \frac{2}{21} a^{2} + \frac{13}{28} a + \frac{4}{21}$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{42} a^{7} - \frac{1}{42} a^{6} + \frac{2}{21} a^{5} - \frac{3}{14} a^{4} - \frac{13}{42} a^{3} + \frac{5}{21} a^{2} + \frac{3}{14} a + \frac{5}{21} \),  \( \frac{5}{84} a^{7} - \frac{5}{12} a^{6} + \frac{2}{3} a^{5} + \frac{3}{4} a^{4} - \frac{29}{12} a^{3} + \frac{2}{3} a^{2} + \frac{7}{4} a - \frac{37}{21} \),  \( \frac{1}{84} a^{7} - \frac{1}{12} a^{6} + \frac{1}{3} a^{5} - \frac{1}{4} a^{4} - \frac{13}{12} a^{3} + \frac{4}{3} a^{2} - \frac{1}{4} a - \frac{62}{21} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 66.4798351838 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2:C_3$ (as 8T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 10 conjugacy class representatives for $C_2^4:C_6$
Character table for $C_2^4:C_6$

Intermediate fields

\(\Q(\sqrt{21}) \)

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 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$