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Properties

Label 8.0.104658502321...6249.3
Degree $8$
Signature $[0, 4]$
Discriminant $53^{6}\cdot 241^{7}$
Root discriminant $2384.91$
Ramified primes $53, 241$
Class number $174072592$ (GRH)
Class group $[4, 43518148]$ (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![1810372179920, 13294112308, -391513312, 48408231, -935391, 32742, 1582, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 1582*x^6 + 32742*x^5 - 935391*x^4 + 48408231*x^3 - 391513312*x^2 + 13294112308*x + 1810372179920)
 
gp: K = bnfinit(x^8 - x^7 + 1582*x^6 + 32742*x^5 - 935391*x^4 + 48408231*x^3 - 391513312*x^2 + 13294112308*x + 1810372179920, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 1582 x^{6} + 32742 x^{5} - 935391 x^{4} + 48408231 x^{3} - 391513312 x^{2} + 13294112308 x + 1810372179920 \)

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:  \(1046585023211040080984056249=53^{6}\cdot 241^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2384.91$
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:  $53, 241$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(12773=53\cdot 241\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{72} a^{4} - \frac{1}{12} a^{3} - \frac{5}{72} a^{2} - \frac{5}{12} a - \frac{4}{9}$, $\frac{1}{3240} a^{5} - \frac{1}{648} a^{4} + \frac{53}{648} a^{3} + \frac{5}{648} a^{2} + \frac{137}{1620} a + \frac{13}{81}$, $\frac{1}{194400} a^{6} + \frac{23}{194400} a^{5} - \frac{137}{38880} a^{4} + \frac{1273}{38880} a^{3} + \frac{3821}{48600} a^{2} - \frac{2947}{48600} a - \frac{979}{2430}$, $\frac{1}{26874292201232400000} a^{7} + \frac{12363266882359}{6718573050308100000} a^{6} - \frac{63633057920327}{1493016233401800000} a^{5} + \frac{126872364989921}{74650811670090000} a^{4} + \frac{49994313062100443}{8958097400410800000} a^{3} - \frac{23791559081799583}{2239524350102700000} a^{2} + \frac{257055543507934859}{6718573050308100000} a - \frac{151341145119688877}{335928652515405000}$

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

Class group and class number

$C_{4}\times C_{43518148}$, which has order $174072592$ (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:  \( 1521894.52814 \) (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

\(\Q(\sqrt{241}) \), 4.4.39319036489.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$53$53.4.3.4$x^{4} + 424$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.4$x^{4} + 424$$4$$1$$3$$C_4$$[\ ]_{4}$
241Data not computed