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Properties

Label 8.0.604673516457...616.46
Degree $8$
Signature $[0, 4]$
Discriminant $2^{24}\cdot 1201^{7}$
Root discriminant $3959.95$
Ramified primes $2, 1201$
Class number $235520000$ (GRH)
Class group $[4, 4, 8, 40, 46000]$ (GRH)
Galois group $((C_8 : C_2):C_2):C_2$ (as 8T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27936856129, 0, 1330213188, 0, 4811206, 0, 4804, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4804*x^6 + 4811206*x^4 + 1330213188*x^2 + 27936856129)
 
gp: K = bnfinit(x^8 + 4804*x^6 + 4811206*x^4 + 1330213188*x^2 + 27936856129, 1)
 

Normalized defining polynomial

\( x^{8} + 4804 x^{6} + 4811206 x^{4} + 1330213188 x^{2} + 27936856129 \)

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:  \(60467351645799489443637231616=2^{24}\cdot 1201^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3959.95$
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:  $2, 1201$
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 \(a\))

$1$, $a$, $\frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{4} - \frac{1}{8} a^{2} - \frac{3}{16}$, $\frac{1}{224} a^{5} - \frac{1}{32} a^{4} + \frac{9}{112} a^{3} - \frac{1}{16} a^{2} - \frac{47}{224} a - \frac{1}{32}$, $\frac{1}{56966740096} a^{6} - \frac{1767627333}{56966740096} a^{4} - \frac{1}{8} a^{3} - \frac{6780058941}{56966740096} a^{2} - \frac{1}{8} a + \frac{1803463791}{8138105728}$, $\frac{1}{39250083926144} a^{7} + \frac{1284162315}{39250083926144} a^{5} - \frac{1062699277149}{39250083926144} a^{3} - \frac{1}{8} a^{2} + \frac{5879181213209}{39250083926144} a - \frac{1}{8}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{8}\times C_{40}\times C_{46000}$, which has order $235520000$ (assuming GRH)

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:  \( \frac{12171}{28483370048} a^{6} + \frac{53169437}{28483370048} a^{4} + \frac{35390291961}{28483370048} a^{2} + \frac{107536342113}{4069052864} \),  \( \frac{5095}{28483370048} a^{6} + \frac{24305417}{28483370048} a^{4} + \frac{23729670109}{28483370048} a^{2} + \frac{873284171757}{4069052864} \),  \( \frac{3}{35515424} a^{6} + \frac{14747}{35515424} a^{4} + \frac{14600557}{35515424} a^{2} + \frac{543826979}{5073632} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2043.83946824 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 8T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$
Character table for $((C_8 : C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{2402}) \), 4.4.3547798734848.2

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
1201Data not computed