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Properties

Label 8.0.783242835309...6161.2
Degree $8$
Signature $[0, 4]$
Discriminant $97^{6}\cdot 313^{6}$
Root discriminant $2300.05$
Ramified primes $97, 313$
Class number $107809928$ (GRH)
Class group $[2, 7342, 7342]$ (GRH)
Galois group $Q_8$ (as 8T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9286969469184, 598996548288, 9794653488, 123908628, 17412064, -55449, 5671, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 5671*x^6 - 55449*x^5 + 17412064*x^4 + 123908628*x^3 + 9794653488*x^2 + 598996548288*x + 9286969469184)
 
gp: K = bnfinit(x^8 - 3*x^7 + 5671*x^6 - 55449*x^5 + 17412064*x^4 + 123908628*x^3 + 9794653488*x^2 + 598996548288*x + 9286969469184, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 5671 x^{6} - 55449 x^{5} + 17412064 x^{4} + 123908628 x^{3} + 9794653488 x^{2} + 598996548288 x + 9286969469184 \)

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:  \(783242835309646294823246161=97^{6}\cdot 313^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2300.05$
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:  $97, 313$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{24} a^{4} - \frac{1}{12} a^{3} - \frac{1}{24} a^{2} + \frac{1}{12} a$, $\frac{1}{264} a^{5} + \frac{1}{132} a^{4} - \frac{17}{264} a^{3} - \frac{31}{132} a^{2} + \frac{7}{66} a$, $\frac{1}{57024} a^{6} + \frac{31}{19008} a^{5} + \frac{77}{5184} a^{4} + \frac{989}{19008} a^{3} + \frac{335}{7128} a^{2} - \frac{335}{1584} a - \frac{17}{36}$, $\frac{1}{8320573718603747030605824} a^{7} - \frac{2700551272009805359}{924508190955971892289536} a^{6} - \frac{10305343037766634616285}{8320573718603747030605824} a^{5} + \frac{29209831911075554572969}{2773524572867915676868608} a^{4} - \frac{68999113892401473275771}{2080143429650936757651456} a^{3} + \frac{74646567696211264697627}{693381143216978919217152} a^{2} + \frac{6367135273481838626725}{28890880967374121634048} a + \frac{75170307168704311385}{437740620717789721728}$

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

Class group and class number

$C_{2}\times C_{7342}\times C_{7342}$, which has order $107809928$ (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:  \( 522483.769592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8$ (as 8T5):

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 $Q_8$
Character table for $Q_8$

Intermediate fields

\(\Q(\sqrt{30361}) \), \(\Q(\sqrt{313}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{97}, \sqrt{313})\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$97$97.4.3.2$x^{4} - 2425$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.3.2$x^{4} - 2425$$4$$1$$3$$C_4$$[\ ]_{4}$
313Data not computed