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Properties

Label 8.0.566660933138881.4
Degree $8$
Signature $[0, 4]$
Discriminant $7^{4}\cdot 17^{4}\cdot 41^{4}$
Root discriminant $69.85$
Ramified primes $7, 17, 41$
Class number $2730$
Class group $[2730]$
Galois group $C_2^3$ (as 8T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8192, -704, 1496, -1534, 643, 146, -44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 44*x^6 + 146*x^5 + 643*x^4 - 1534*x^3 + 1496*x^2 - 704*x + 8192)
 
gp: K = bnfinit(x^8 - 4*x^7 - 44*x^6 + 146*x^5 + 643*x^4 - 1534*x^3 + 1496*x^2 - 704*x + 8192, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} - 44 x^{6} + 146 x^{5} + 643 x^{4} - 1534 x^{3} + 1496 x^{2} - 704 x + 8192 \)

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:  \(566660933138881=7^{4}\cdot 17^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.85$
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:  $7, 17, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4879=7\cdot 17\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4879}(288,·)$, $\chi_{4879}(1,·)$, $\chi_{4879}(2787,·)$, $\chi_{4879}(2500,·)$, $\chi_{4879}(2379,·)$, $\chi_{4879}(2092,·)$, $\chi_{4879}(4878,·)$, $\chi_{4879}(4591,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{24} a^{4} - \frac{1}{12} a^{3} - \frac{5}{24} a^{2} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{24} a^{5} - \frac{1}{24} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{576} a^{6} - \frac{1}{192} a^{5} - \frac{7}{576} a^{4} + \frac{19}{576} a^{3} + \frac{47}{288} a^{2} - \frac{13}{72} a + \frac{4}{9}$, $\frac{1}{275904} a^{7} + \frac{59}{68976} a^{6} + \frac{737}{68976} a^{5} - \frac{2543}{137952} a^{4} - \frac{1861}{30656} a^{3} + \frac{2939}{45984} a^{2} + \frac{895}{3832} a + \frac{1181}{4311}$

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

Class group and class number

$C_{2730}$, which has order $2730$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 777.260380927 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3$ (as 8T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 8
The 8 conjugacy class representatives for $C_2^3$
Character table for $C_2^3$

Intermediate fields

\(\Q(\sqrt{-4879}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{697}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-287}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-7}, \sqrt{697})\), \(\Q(\sqrt{41}, \sqrt{-119})\), \(\Q(\sqrt{17}, \sqrt{-287})\), \(\Q(\sqrt{-7}, \sqrt{41})\), \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\sqrt{17}, \sqrt{41})\), \(\Q(\sqrt{-119}, \sqrt{-287})\)

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$