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Properties

Label 8.0.456566384392...8497.1
Degree $8$
Signature $[0, 4]$
Discriminant $97^{6}\cdot 353^{5}$
Root discriminant $1209.03$
Ramified primes $97, 353$
Class number $40589888$ (GRH)
Class group $[2, 2, 4, 2536868]$ (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![601602283008, 30776452128, 5881984412, 91668357, 14862865, 17246, 8530, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 8530*x^6 + 17246*x^5 + 14862865*x^4 + 91668357*x^3 + 5881984412*x^2 + 30776452128*x + 601602283008)
 
gp: K = bnfinit(x^8 - 3*x^7 + 8530*x^6 + 17246*x^5 + 14862865*x^4 + 91668357*x^3 + 5881984412*x^2 + 30776452128*x + 601602283008, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 8530 x^{6} + 17246 x^{5} + 14862865 x^{4} + 91668357 x^{3} + 5881984412 x^{2} + 30776452128 x + 601602283008 \)

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:  \(4565663843921795982558497=97^{6}\cdot 353^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1209.03$
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:  $97, 353$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{96} a^{6} + \frac{1}{48} a^{5} - \frac{1}{24} a^{4} - \frac{1}{16} a^{3} + \frac{19}{96} a^{2} + \frac{5}{24} a$, $\frac{1}{617685673236398039647869666816} a^{7} + \frac{785354649942647654796147523}{154421418309099509911967416704} a^{6} - \frac{2879267434740091901402815391}{102947612206066339941311611136} a^{5} + \frac{1694663974931629057315312847}{77210709154549754955983708352} a^{4} + \frac{27809351745360841613789084569}{617685673236398039647869666816} a^{3} + \frac{19760566073638074901248403231}{154421418309099509911967416704} a^{2} - \frac{3203241558588865454310292807}{19302677288637438738995927088} a + \frac{54369922567023101629860749}{402139110179946640395748481}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{2536868}$, which has order $40589888$ (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:  \( 11531.8652003 \) (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

\(\Q(\sqrt{97}) \), 4.4.3321377.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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}$
353Data not computed