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Properties

Label 8.0.4372515625.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{6}\cdot 23^{4}$
Root discriminant $16.04$
Ramified primes $5, 23$
Class number $3$
Class group $[3]$
Galois group $C_4\times C_2$ (as 8T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, -216, -180, 66, 19, 11, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 5*x^6 + 11*x^5 + 19*x^4 + 66*x^3 - 180*x^2 - 216*x + 1296)
 
gp: K = bnfinit(x^8 - x^7 - 5*x^6 + 11*x^5 + 19*x^4 + 66*x^3 - 180*x^2 - 216*x + 1296, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 5 x^{6} + 11 x^{5} + 19 x^{4} + 66 x^{3} - 180 x^{2} - 216 x + 1296 \)

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:  \(4372515625=5^{6}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.04$
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:  $5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(68,·)$, $\chi_{115}(47,·)$, $\chi_{115}(114,·)$, $\chi_{115}(22,·)$, $\chi_{115}(24,·)$, $\chi_{115}(91,·)$, $\chi_{115}(93,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{114} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{8}{19}$, $\frac{1}{684} a^{6} - \frac{1}{684} a^{5} + \frac{13}{36} a^{4} - \frac{7}{36} a^{3} + \frac{1}{36} a^{2} + \frac{11}{114} a - \frac{5}{19}$, $\frac{1}{4104} a^{7} - \frac{1}{4104} a^{6} - \frac{5}{4104} a^{5} - \frac{79}{216} a^{4} + \frac{1}{216} a^{3} + \frac{11}{684} a^{2} - \frac{5}{114} a - \frac{1}{19}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{5}{4104} a^{7} - \frac{25}{4104} a^{6} + \frac{55}{4104} a^{5} + \frac{5}{216} a^{4} - \frac{11}{216} a^{3} - \frac{25}{114} a^{2} - \frac{5}{19} a + \frac{30}{19} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39.3022218671 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4$ (as 8T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 8
The 8 conjugacy class representatives for $C_4\times C_2$
Character table for $C_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\zeta_{5})\), 4.4.66125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$