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Properties

Label 8.8.79803075463713.1
Degree $8$
Signature $[8, 0]$
Discriminant $3^{4}\cdot 7^{4}\cdot 17^{7}$
Root discriminant $54.67$
Ramified primes $3, 7, 17$
Class number $2$
Class group $[2]$
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![-7649, 28309, -12590, -3325, 2055, 91, -92, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 92*x^6 + 91*x^5 + 2055*x^4 - 3325*x^3 - 12590*x^2 + 28309*x - 7649)
 
gp: K = bnfinit(x^8 - x^7 - 92*x^6 + 91*x^5 + 2055*x^4 - 3325*x^3 - 12590*x^2 + 28309*x - 7649, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 92 x^{6} + 91 x^{5} + 2055 x^{4} - 3325 x^{3} - 12590 x^{2} + 28309 x - 7649 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(79803075463713=3^{4}\cdot 7^{4}\cdot 17^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.67$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(64,·)$, $\chi_{357}(1,·)$, $\chi_{357}(230,·)$, $\chi_{357}(104,·)$, $\chi_{357}(169,·)$, $\chi_{357}(106,·)$, $\chi_{357}(83,·)$, $\chi_{357}(314,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} - \frac{2}{13} a^{5} - \frac{5}{13} a^{4} + \frac{4}{13} a^{3} + \frac{1}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{4921913113} a^{7} + \frac{25771957}{4921913113} a^{6} + \frac{1306434816}{4921913113} a^{5} - \frac{1965453524}{4921913113} a^{4} - \frac{2339998991}{4921913113} a^{3} - \frac{59926078}{4921913113} a^{2} + \frac{297003165}{4921913113} a + \frac{541288525}{4921913113}$

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

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11333.9362425 \)
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

\(\Q(\sqrt{17}) \), 4.4.4913.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }$ R ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$7$7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$