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Properties

Label 8.0.632857188437...0593.2
Degree $8$
Signature $[0, 4]$
Discriminant $401^{6}\cdot 433^{5}$
Root discriminant $3982.57$
Ramified primes $401, 433$
Class number $112921200$ (GRH)
Class group $[3, 37640400]$ (GRH)
Galois group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79318672, 0, 87184983, 0, 1501603, 0, 4469, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4469*x^6 + 1501603*x^4 + 87184983*x^2 + 79318672)
 
gp: K = bnfinit(x^8 + 4469*x^6 + 1501603*x^4 + 87184983*x^2 + 79318672, 1)
 

Normalized defining polynomial

\( x^{8} + 4469 x^{6} + 1501603 x^{4} + 87184983 x^{2} + 79318672 \)

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:  \(63285718843729333630079120593=401^{6}\cdot 433^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3982.57$
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:  $401, 433$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{9999862784} a^{6} - \frac{162467893}{4999931392} a^{4} + \frac{974088697}{9999862784} a^{2} - \frac{1}{2} a + \frac{141393143}{624991424}$, $\frac{1}{2139970635776} a^{7} - \frac{1}{19999725568} a^{6} + \frac{28587137611}{1069985317888} a^{5} + \frac{162467893}{9999862784} a^{4} - \frac{116524299015}{2139970635776} a^{3} + \frac{4025842695}{19999725568} a^{2} + \frac{62953031255}{133748164736} a + \frac{483598281}{1249982848}$

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

Class group and class number

$C_{3}\times C_{37640400}$, which has order $112921200$ (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:  \( \frac{1505549}{2499965696} a^{6} + \frac{3269721167}{1249982848} a^{4} + \frac{1428367607461}{2499965696} a^{2} - \frac{5757538679621}{156247856} \),  \( \frac{1187257}{4999931392} a^{6} + \frac{2459915315}{2499965696} a^{4} + \frac{121991620849}{4999931392} a^{2} - \frac{1049245847137}{312495712} \),  \( \frac{388460159619}{624991424} a^{6} + \frac{69982550150473}{312495712} a^{4} + \frac{8241230540866203}{624991424} a^{2} + \frac{468707648420345}{39061964} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 247134.284786 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_4$ (as 8T30):

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 $(((C_4 \times C_2): C_2):C_2):C_2$
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{173633}) \), 4.4.12089515894289.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
433Data not computed