↑

Properties

Label 8.0.523550526046...4889.2
Degree $8$
Signature $[0, 4]$
Discriminant $313^{6}\cdot 421^{6}$
Root discriminant $6916.24$
Ramified primes $313, 421$
Class number $1174056960$ (GRH)
Class group $[2, 2, 8, 8, 4586160]$ (GRH)
Galois group $C_4\times C_2$ (as 8T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2101736448182448, -19047412684944, -81144892584, -3890248614, 210268857, 1813972, 24662, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 24662*x^6 + 1813972*x^5 + 210268857*x^4 - 3890248614*x^3 - 81144892584*x^2 - 19047412684944*x + 2101736448182448)
 
gp: K = bnfinit(x^8 - x^7 + 24662*x^6 + 1813972*x^5 + 210268857*x^4 - 3890248614*x^3 - 81144892584*x^2 - 19047412684944*x + 2101736448182448, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 24662 x^{6} + 1813972 x^{5} + 210268857 x^{4} - 3890248614 x^{3} - 81144892584 x^{2} - 19047412684944 x + 2101736448182448 \)

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:  \(5235505260467902422785504364889=313^{6}\cdot 421^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6916.24$
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:  $313, 421$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(131773=313\cdot 421\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{4} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{126} a^{5} - \frac{1}{126} a^{4} + \frac{4}{63} a^{3} - \frac{25}{63} a^{2} + \frac{5}{42} a$, $\frac{1}{342798418116} a^{6} - \frac{42951323}{31163492556} a^{5} - \frac{7711645259}{171399209058} a^{4} - \frac{10680639440}{85699604529} a^{3} - \frac{8050112899}{38088713124} a^{2} - \frac{1022623421}{2720622366} a + \frac{787056}{1962931}$, $\frac{1}{10739520084982021737615288} a^{7} - \frac{9626556949927}{10739520084982021737615288} a^{6} - \frac{1962635655018443189713}{2684880021245505434403822} a^{5} - \frac{59098190935279858046182}{1342440010622752717201911} a^{4} - \frac{207074015874471117392533}{3579840028327340579205096} a^{3} - \frac{4107884655193571139697}{42617143194373102133394} a^{2} + \frac{78384217609539023689}{289912538737231987302} a + \frac{7654632976960201}{488068247032377083}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{4586160}$, which has order $1174056960$ (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:  \( 23736.3707289 \) (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

\(\Q(\sqrt{421}) \), \(\Q(\sqrt{131773}) \), \(\Q(\sqrt{313}) \), \(\Q(\sqrt{313}, \sqrt{421})\), 4.0.2288122649786917.1, 4.0.2288122649786917.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
313Data not computed
421Data not computed