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Properties

Label 8.0.137744855669...9193.5
Degree $8$
Signature $[0, 4]$
Discriminant $13^{6}\cdot 433^{7}$
Root discriminant $1387.98$
Ramified primes $13, 433$
Class number $14123664$ (GRH)
Class group $[12, 1176972]$ (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![56703181824, 714289536, -80790688, -3678248, -134234, -16515, 677, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 677*x^6 - 16515*x^5 - 134234*x^4 - 3678248*x^3 - 80790688*x^2 + 714289536*x + 56703181824)
 
gp: K = bnfinit(x^8 - x^7 + 677*x^6 - 16515*x^5 - 134234*x^4 - 3678248*x^3 - 80790688*x^2 + 714289536*x + 56703181824, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 677 x^{6} - 16515 x^{5} - 134234 x^{4} - 3678248 x^{3} - 80790688 x^{2} + 714289536 x + 56703181824 \)

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:  \(13774485566948763382269193=13^{6}\cdot 433^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1387.98$
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:  $13, 433$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5629=13\cdot 433\)
Dirichlet character group:    $\lbrace$$\chi_{5629}(1,·)$, $\chi_{5629}(2244,·)$, $\chi_{5629}(3749,·)$, $\chi_{5629}(3210,·)$, $\chi_{5629}(148,·)$, $\chi_{5629}(3030,·)$, $\chi_{5629}(5017,·)$, $\chi_{5629}(5117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a$, $\frac{1}{24} a^{4} + \frac{5}{24} a^{2} - \frac{1}{4} a$, $\frac{1}{1728} a^{5} - \frac{7}{432} a^{4} - \frac{31}{1728} a^{3} + \frac{137}{864} a^{2} - \frac{7}{24} a - \frac{1}{3}$, $\frac{1}{5184} a^{6} + \frac{1}{5184} a^{5} - \frac{17}{1728} a^{4} + \frac{95}{5184} a^{3} - \frac{563}{2592} a^{2} + \frac{31}{72} a - \frac{2}{9}$, $\frac{1}{3548049534623683584} a^{7} - \frac{14243199033077}{394227726069298176} a^{6} - \frac{667738795869295}{3548049534623683584} a^{5} + \frac{1112988143712161}{3548049534623683584} a^{4} + \frac{3829624215612023}{591341589103947264} a^{3} - \frac{10693634800976837}{55438273978495056} a^{2} + \frac{3368682620800099}{12319616439665568} a + \frac{76643149585919}{1539952054958196}$

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

Class group and class number

$C_{12}\times C_{1176972}$, which has order $14123664$ (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:  \( 662848.605206 \) (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

\(\Q(\sqrt{433}) \), 4.4.13719882553.2

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. 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
$13$13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
433Data not computed