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Properties

Label 8.0.677922439794...3761.2
Degree $8$
Signature $[0, 4]$
Discriminant $113^{4}\cdot 401^{6}$
Root discriminant $952.57$
Ramified primes $113, 401$
Class number $63201960$ (GRH)
Class group $[838, 75420]$ (GRH)
Galois group $D_4$ (as 8T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![483032484, 0, -3696749, 0, 43247, 0, 473, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 473*x^6 + 43247*x^4 - 3696749*x^2 + 483032484)
 
gp: K = bnfinit(x^8 + 473*x^6 + 43247*x^4 - 3696749*x^2 + 483032484, 1)
 

Normalized defining polynomial

\( x^{8} + 473 x^{6} + 43247 x^{4} - 3696749 x^{2} + 483032484 \)

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:  \(677922439794791223113761=113^{4}\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $952.57$
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:  $113, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{112} a^{4} - \frac{13}{112} a^{2} - \frac{1}{2} a + \frac{9}{28}$, $\frac{1}{22176} a^{5} - \frac{1}{224} a^{4} + \frac{235}{2464} a^{3} - \frac{43}{224} a^{2} + \frac{23}{5544} a + \frac{19}{56}$, $\frac{1}{2483712} a^{6} - \frac{163}{137984} a^{4} + \frac{104537}{2483712} a^{2} + \frac{717}{6272}$, $\frac{1}{551384064} a^{7} - \frac{1}{4967424} a^{6} - \frac{3371}{275692032} a^{5} - \frac{1069}{275968} a^{4} - \frac{50172487}{551384064} a^{3} - \frac{1058105}{4967424} a^{2} - \frac{60867545}{137846016} a - \frac{2733}{12544}$

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

Class group and class number

$C_{838}\times C_{75420}$, which has order $63201960$ (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:  \( \frac{1}{2813184} a^{7} + \frac{181}{1406592} a^{5} + \frac{17273}{2813184} a^{3} - \frac{1275641}{703296} a - 20 \),  \( \frac{365}{68923008} a^{7} + \frac{11899}{3132864} a^{5} + \frac{23807125}{68923008} a^{3} - \frac{1399593157}{17230752} a - 776 \),  \( \frac{1409}{19692288} a^{7} + \frac{98741}{9846144} a^{5} - \frac{17860103}{19692288} a^{3} + \frac{454419335}{4923072} a \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8223.52656633 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4$ (as 8T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8
The 5 conjugacy class representatives for $D_4$
Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{113}) \), \(\Q(\sqrt{401}) \), \(\Q(\sqrt{45313}) \), \(\Q(\sqrt{113}, \sqrt{401})\), 4.0.7286375713.1 x2, 4.0.823360455569.1 x2

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

Sibling fields

Degree 4 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$113$113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed