↑

Properties

Label 8.0.249694239268...8649.2
Degree $8$
Signature $[0, 4]$
Discriminant $157^{6}\cdot 401^{7}$
Root discriminant $8407.68$
Ramified primes $157, 401$
Class number $1047824020$ (GRH)
Class group $[1047824020]$ (GRH)
Galois group $C_8$ (as 8T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![816267426117248, -19422288907392, 76480434632, -2848018388, 66210090, -405267, 7845, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 7845*x^6 - 405267*x^5 + 66210090*x^4 - 2848018388*x^3 + 76480434632*x^2 - 19422288907392*x + 816267426117248)
 
gp: K = bnfinit(x^8 - x^7 + 7845*x^6 - 405267*x^5 + 66210090*x^4 - 2848018388*x^3 + 76480434632*x^2 - 19422288907392*x + 816267426117248, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 7845 x^{6} - 405267 x^{5} + 66210090 x^{4} - 2848018388 x^{3} + 76480434632 x^{2} - 19422288907392 x + 816267426117248 \)

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:  \(24969423926841105551040628528649=157^{6}\cdot 401^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8407.68$
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:  $157, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(62957=157\cdot 401\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{112} a^{4} - \frac{1}{56} a^{3} - \frac{5}{112} a^{2} + \frac{3}{56} a - \frac{1}{2}$, $\frac{1}{224} a^{5} - \frac{9}{224} a^{3} - \frac{1}{56} a^{2} + \frac{17}{56} a - \frac{1}{2}$, $\frac{1}{448} a^{6} - \frac{1}{448} a^{5} - \frac{1}{448} a^{4} + \frac{101}{448} a^{3} + \frac{1}{14} a^{2} + \frac{51}{112} a + \frac{1}{4}$, $\frac{1}{128212994542769388552820077727232} a^{7} - \frac{59230773832074265522390925511}{128212994542769388552820077727232} a^{6} - \frac{261904072593017254380292379153}{128212994542769388552820077727232} a^{5} + \frac{340786335959616607622401853347}{128212994542769388552820077727232} a^{4} - \frac{2154015814609909284757527026689}{16026624317846173569102509715904} a^{3} + \frac{1769042848424107681764043309203}{32053248635692347138205019431808} a^{2} + \frac{34017285685844103111738052251}{71547429990384703433493346946} a + \frac{8765050854630074944198510497}{40884245708791259104853341112}$

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

Class group and class number

$C_{1047824020}$, which has order $1047824020$ (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:  \( 632238.89986 \) (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

\(\Q(\sqrt{401}) \), 4.4.1589397123449.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$157$157.8.6.4$x^{8} + 2669 x^{4} + 5324184$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
401Data not computed