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Properties

Label 8.0.190821431004...8249.1
Degree $8$
Signature $[0, 4]$
Discriminant $401^{7}\cdot 409^{5}$
Root discriminant $8129.77$
Ramified primes $401, 409$
Class number $2468464960$ (GRH)
Class group $[4, 4, 154279060]$ (GRH)
Galois group $C_4\wr C_2$ (as 8T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63545277135872, -4749902474240, 339739632560, -63041308, 247972964, -163063, 40727, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 40727*x^6 - 163063*x^5 + 247972964*x^4 - 63041308*x^3 + 339739632560*x^2 - 4749902474240*x + 63545277135872)
 
gp: K = bnfinit(x^8 - x^7 + 40727*x^6 - 163063*x^5 + 247972964*x^4 - 63041308*x^3 + 339739632560*x^2 - 4749902474240*x + 63545277135872, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 40727 x^{6} - 163063 x^{5} + 247972964 x^{4} - 63041308 x^{3} + 339739632560 x^{2} - 4749902474240 x + 63545277135872 \)

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:  \(19082143100442103109705931158249=401^{7}\cdot 409^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8129.77$
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:  $401, 409$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{5} + \frac{1}{32} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{1024} a^{6} + \frac{5}{512} a^{5} - \frac{3}{1024} a^{4} - \frac{5}{128} a^{3} - \frac{47}{256} a^{2} + \frac{21}{64} a + \frac{3}{8}$, $\frac{1}{6104354687844949814771885793028096} a^{7} - \frac{466858705480374634329588307939}{1526088671961237453692971448257024} a^{6} - \frac{52612324333742819801311462263775}{6104354687844949814771885793028096} a^{5} + \frac{51774409463168838895292560109709}{3052177343922474907385942896514048} a^{4} + \frac{20267315293882440996612015359949}{1526088671961237453692971448257024} a^{3} - \frac{2431935179283726463616236040529}{763044335980618726846485724128512} a^{2} + \frac{409827661460535055715208518967}{10040057052376562195348496370112} a - \frac{1360295704005187778427203073543}{3406447928484905030564668411288}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{154279060}$, which has order $2468464960$ (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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 77242.5264237 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 8T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.4.26372811209.1

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

Sibling fields

Galois closure: data not computed
Degree 8 sibling: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
401Data not computed
409Data not computed