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Properties

Label 8.0.168630239892...0609.4
Degree $8$
Signature $[0, 4]$
Discriminant $193^{6}\cdot 239^{4}$
Root discriminant $800.51$
Ramified primes $193, 239$
Class number $11172000$ (GRH)
Class group $[10, 70, 15960]$ (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![375083136, -200704608, 35342184, -1755674, 25657, 4924, -290, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 290*x^6 + 4924*x^5 + 25657*x^4 - 1755674*x^3 + 35342184*x^2 - 200704608*x + 375083136)
 
gp: K = bnfinit(x^8 - 2*x^7 - 290*x^6 + 4924*x^5 + 25657*x^4 - 1755674*x^3 + 35342184*x^2 - 200704608*x + 375083136, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 290 x^{6} + 4924 x^{5} + 25657 x^{4} - 1755674 x^{3} + 35342184 x^{2} - 200704608 x + 375083136 \)

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:  \(168630239892922523260609=193^{6}\cdot 239^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $800.51$
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:  $193, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{144} a^{4} + \frac{1}{24} a^{3} - \frac{1}{144} a^{2} + \frac{7}{24} a - \frac{1}{2}$, $\frac{1}{432} a^{5} + \frac{1}{432} a^{4} - \frac{31}{432} a^{3} - \frac{97}{432} a^{2} - \frac{23}{72} a - \frac{1}{2}$, $\frac{1}{1238976} a^{6} + \frac{385}{412992} a^{5} - \frac{1043}{1238976} a^{4} - \frac{2561}{412992} a^{3} + \frac{36845}{619488} a^{2} + \frac{22103}{51624} a + \frac{20}{717}$, $\frac{1}{1379470922496} a^{7} + \frac{19427}{76637273472} a^{6} + \frac{489481}{16822816128} a^{5} - \frac{74808347}{57477955104} a^{4} + \frac{79924148749}{1379470922496} a^{3} - \frac{1362658805}{5607605376} a^{2} + \frac{756624769}{3193219728} a - \frac{1387565}{5722616}$

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

Class group and class number

$C_{10}\times C_{70}\times C_{15960}$, which has order $11172000$ (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:  \( 55944.6653583 \) (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

\(\Q(\sqrt{-239}) \), \(\Q(\sqrt{193}) \), \(\Q(\sqrt{-46127}) \), \(\Q(\sqrt{193}, \sqrt{-239})\), 4.2.1718184623.1 x2, 4.0.410646124897.1 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$193$193.4.3.1$x^{4} - 193$$4$$1$$3$$C_4$$[\ ]_{4}$
193.4.3.1$x^{4} - 193$$4$$1$$3$$C_4$$[\ ]_{4}$
239Data not computed