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Properties

Label 8.0.171043633626...8313.5
Degree $8$
Signature $[0, 4]$
Discriminant $337^{7}\cdot 389^{6}$
Root discriminant $14{,}260.62$
Ramified primes $337, 389$
Class number $1064272896$ (GRH)
Class group $[2, 2, 2, 4, 12, 2771544]$ (GRH)
Galois group $C_8$ (as 8T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18516155826524688, -275329019071512, 1419762761784, -30103158100, 230733571, -1696000, 16366, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 16366*x^6 - 1696000*x^5 + 230733571*x^4 - 30103158100*x^3 + 1419762761784*x^2 - 275329019071512*x + 18516155826524688)
 
gp: K = bnfinit(x^8 - x^7 + 16366*x^6 - 1696000*x^5 + 230733571*x^4 - 30103158100*x^3 + 1419762761784*x^2 - 275329019071512*x + 18516155826524688, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 16366 x^{6} - 1696000 x^{5} + 230733571 x^{4} - 30103158100 x^{3} + 1419762761784 x^{2} - 275329019071512 x + 18516155826524688 \)

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:  \(1710436336269048136563476916048313=337^{7}\cdot 389^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14{,}260.62$
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:  $337, 389$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(131093=337\cdot 389\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{468} a^{6} + \frac{7}{156} a^{5} + \frac{10}{117} a^{4} - \frac{1}{39} a^{3} - \frac{149}{468} a^{2} - \frac{7}{26} a + \frac{3}{13}$, $\frac{1}{4230435275857041082197025645276702632} a^{7} + \frac{580402235081556850649605841081795}{1410145091952347027399008548425567544} a^{6} + \frac{33383094071963699582384690451123667}{1057608818964260270549256411319175658} a^{5} - \frac{32994287098552148444099780282203771}{352536272988086756849752137106391886} a^{4} - \frac{235079346557864812176255438814784837}{4230435275857041082197025645276702632} a^{3} - \frac{180153491846054175956077648048846625}{705072545976173513699504274212783772} a^{2} + \frac{52475491783021442642346125522331}{477691426813125686788282028599447} a + \frac{5782274494459581547795560551826483}{19585348499338153158319563172577327}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{2771544}$, which has order $1064272896$ (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:  \( 676331.383104 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8$ (as 8T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 8
The 8 conjugacy class representatives for $C_8$
Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{337}) \), 4.4.5791471256713.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
337Data not computed
389Data not computed