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Properties

Label 8.0.5922408960000.16
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4}$
Root discriminant $39.50$
Ramified primes $2, 3, 5, 13$
Class number $128$
Class group $[4, 4, 8]$
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![21711, -5052, 5356, -616, 329, -28, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 329*x^4 - 616*x^3 + 5356*x^2 - 5052*x + 21711)
 
gp: K = bnfinit(x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 329*x^4 - 616*x^3 + 5356*x^2 - 5052*x + 21711, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 329 x^{4} - 616 x^{3} + 5356 x^{2} - 5052 x + 21711 \)

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:  \(5922408960000=2^{12}\cdot 3^{4}\cdot 5^{4}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.50$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1560=2^{3}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(389,·)$, $\chi_{1560}(961,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(1429,·)$, $\chi_{1560}(521,·)$, $\chi_{1560}(989,·)$, $\chi_{1560}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{5} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{1290} a^{6} - \frac{1}{430} a^{5} - \frac{31}{645} a^{4} + \frac{1}{10} a^{3} + \frac{299}{645} a^{2} + \frac{209}{430} a - \frac{87}{430}$, $\frac{1}{2952810} a^{7} + \frac{163}{421830} a^{6} + \frac{6827}{210915} a^{5} - \frac{1657}{210915} a^{4} - \frac{57049}{210915} a^{3} + \frac{104684}{210915} a^{2} - \frac{44029}{196854} a + \frac{280807}{984270}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{8}$, which has order $128$

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:  \( \frac{1}{645} a^{6} - \frac{1}{215} a^{5} + \frac{1}{258} a^{4} + \frac{551}{1290} a^{2} - \frac{92}{215} a + \frac{1933}{430} \) (order $6$)
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:  \( 164.641491225 \)
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

\(\Q(\sqrt{-30}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-390}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}, \sqrt{-30})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-30}, \sqrt{-39})\), \(\Q(\sqrt{10}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{10}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{130})\)

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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$