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Properties

Label 8.0.501242099869...9881.1
Degree $8$
Signature $[0, 4]$
Discriminant $101^{6}\cdot 241^{7}$
Root discriminant $3868.17$
Ramified primes $101, 241$
Class number $142737892$ (GRH)
Class group $[11, 12976172]$ (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![176568665184, 101776517136, 16462000842, -151933141, -10382591, 86726, 3028, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 3028*x^6 + 86726*x^5 - 10382591*x^4 - 151933141*x^3 + 16462000842*x^2 + 101776517136*x + 176568665184)
 
gp: K = bnfinit(x^8 - x^7 + 3028*x^6 + 86726*x^5 - 10382591*x^4 - 151933141*x^3 + 16462000842*x^2 + 101776517136*x + 176568665184, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 3028 x^{6} + 86726 x^{5} - 10382591 x^{4} - 151933141 x^{3} + 16462000842 x^{2} + 101776517136 x + 176568665184 \)

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:  \(50124209986911477351431969881=101^{6}\cdot 241^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3868.17$
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:  $101, 241$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(24341=101\cdot 241\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{48} a^{4} - \frac{1}{24} a^{3} - \frac{1}{48} a^{2} - \frac{11}{24} a - \frac{1}{2}$, $\frac{1}{48} a^{5} + \frac{1}{16} a^{3} - \frac{1}{12} a$, $\frac{1}{200448} a^{6} - \frac{629}{66816} a^{5} - \frac{923}{200448} a^{4} - \frac{29}{768} a^{3} - \frac{12643}{100224} a^{2} - \frac{67}{4176} a + \frac{69}{232}$, $\frac{1}{8095244600203302065135616} a^{7} + \frac{901483229516575}{112433952780601417571328} a^{6} + \frac{13074160153772801073331}{2023811150050825516283904} a^{5} - \frac{9812887473183637589069}{1349207433367217010855936} a^{4} - \frac{28301445911666924854493}{8095244600203302065135616} a^{3} - \frac{332502541362651514800919}{1349207433367217010855936} a^{2} - \frac{2324926746134189447159}{56216976390300708785664} a + \frac{520151878154577529269}{3123165355016706043648}$

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

Class group and class number

$C_{11}\times C_{12976172}$, which has order $142737892$ (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:  \( 3252231.33734 \) (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

\(\Q(\sqrt{241}) \), 4.4.142788711721.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$101$101.8.6.4$x^{8} + 808 x^{4} + 275427$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
241Data not computed