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Properties

Label 8.0.35353337448628224.54
Degree $8$
Signature $[0, 4]$
Discriminant $2^{22}\cdot 3^{4}\cdot 101^{4}$
Root discriminant $117.10$
Ramified primes $2, 3, 101$
Class number $10680$ (GRH)
Class group $[10680]$ (GRH)
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![36967006, -1802104, 1836168, -68424, 34953, -892, 302, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 302*x^6 - 892*x^5 + 34953*x^4 - 68424*x^3 + 1836168*x^2 - 1802104*x + 36967006)
 
gp: K = bnfinit(x^8 - 4*x^7 + 302*x^6 - 892*x^5 + 34953*x^4 - 68424*x^3 + 1836168*x^2 - 1802104*x + 36967006, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 302 x^{6} - 892 x^{5} + 34953 x^{4} - 68424 x^{3} + 1836168 x^{2} - 1802104 x + 36967006 \)

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:  \(35353337448628224=2^{22}\cdot 3^{4}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.10$
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:  $2, 3, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4848=2^{4}\cdot 3\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{4848}(1,·)$, $\chi_{4848}(605,·)$, $\chi_{4848}(4241,·)$, $\chi_{4848}(3637,·)$, $\chi_{4848}(1817,·)$, $\chi_{4848}(2425,·)$, $\chi_{4848}(1213,·)$, $\chi_{4848}(3029,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{94241} a^{6} - \frac{3}{94241} a^{5} - \frac{46744}{94241} a^{4} - \frac{748}{94241} a^{3} - \frac{7297}{94241} a^{2} - \frac{39450}{94241} a - \frac{41053}{94241}$, $\frac{1}{9112068049} a^{7} + \frac{48341}{9112068049} a^{6} + \frac{529348403}{9112068049} a^{5} + \frac{972485134}{9112068049} a^{4} - \frac{670316298}{9112068049} a^{3} - \frac{355821330}{9112068049} a^{2} - \frac{2101442554}{9112068049} a - \frac{1171837607}{9112068049}$

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

Class group and class number

$C_{10680}$, which has order $10680$ (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:  \( 19.534360053 \) (assuming GRH)
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

\(\Q(\sqrt{-303}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-606}) \), \(\Q(\sqrt{2}, \sqrt{-303})\), \(\Q(\zeta_{16})^+\), 4.0.188024832.2

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$