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Properties

Label 10.0.71563480803.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 131^{4}$
Root discriminant $12.18$
Ramified primes $3, 131$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 10T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, 3, -9, 1, 2, 2, 0, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - x^8 + 2*x^6 + 2*x^5 + x^4 - 9*x^3 + 3*x^2 + 2*x + 4)
 
gp: K = bnfinit(x^10 - x^9 - x^8 + 2*x^6 + 2*x^5 + x^4 - 9*x^3 + 3*x^2 + 2*x + 4, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} - x^{8} + 2 x^{6} + 2 x^{5} + x^{4} - 9 x^{3} + 3 x^{2} + 2 x + 4 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-71563480803=-\,3^{5}\cdot 131^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.18$
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:  $3, 131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{9}{8} a^{2} + \frac{3}{4} a + \frac{1}{2} \) (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:  \( 114.524734536 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 10T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.17161.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$131$131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$