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Properties

Label 8.0.154691088673...9897.2
Degree $8$
Signature $[0, 4]$
Discriminant $97^{7}\cdot 409^{7}$
Root discriminant $10{,}560.47$
Ramified primes $97, 409$
Class number $1166589912$ (GRH)
Class group $[1166589912]$ (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![321922277979336, -2598191444700, 82900732570, 1567646351, -11857171, 396110, 2480, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 2480*x^6 + 396110*x^5 - 11857171*x^4 + 1567646351*x^3 + 82900732570*x^2 - 2598191444700*x + 321922277979336)
 
gp: K = bnfinit(x^8 - x^7 + 2480*x^6 + 396110*x^5 - 11857171*x^4 + 1567646351*x^3 + 82900732570*x^2 - 2598191444700*x + 321922277979336, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 2480 x^{6} + 396110 x^{5} - 11857171 x^{4} + 1567646351 x^{3} + 82900732570 x^{2} - 2598191444700 x + 321922277979336 \)

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:  \(154691088673896293052793942309897=97^{7}\cdot 409^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10{,}560.47$
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:  $97, 409$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(39673=97\cdot 409\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{3} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{504} a^{4} + \frac{5}{252} a^{3} - \frac{73}{504} a^{2} - \frac{113}{252} a + \frac{2}{7}$, $\frac{1}{2016} a^{5} - \frac{1}{2016} a^{4} + \frac{3}{224} a^{3} - \frac{179}{2016} a^{2} + \frac{227}{504} a + \frac{19}{56}$, $\frac{1}{2201472} a^{6} - \frac{5}{45864} a^{5} + \frac{365}{1100736} a^{4} + \frac{463}{91728} a^{3} + \frac{247705}{2201472} a^{2} - \frac{2453}{10192} a + \frac{4873}{20384}$, $\frac{1}{795194845604104356343296} a^{7} - \frac{16238572258845733}{397597422802052178171648} a^{6} + \frac{65984323849704407375}{397597422802052178171648} a^{5} + \frac{9790416884795885977}{49699677850256522271456} a^{4} - \frac{28085438434528701928043}{795194845604104356343296} a^{3} - \frac{3302625081655234891147}{30584417138619398320896} a^{2} - \frac{507372540349449558461}{22088745711225121009536} a + \frac{2296134580854200465}{14105201603592031296}$

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

Class group and class number

$C_{1166589912}$, which has order $1166589912$ (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:  \( 24968773.6087 \) (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

\(\Q(\sqrt{39673}) \), 4.4.62443196514217.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$97$97.8.7.8$x^{8} + 7578125$$8$$1$$7$$C_8$$[\ ]_{8}$
409Data not computed