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Properties

Label 10.0.47671949454...2896.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{10}\cdot 3^{5}\cdot 19^{4}\cdot 43^{5}$
Root discriminant $73.76$
Ramified primes $2, 3, 19, 43$
Class number $1068$ (GRH)
Class group $[2, 534]$ (GRH)
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![10449, 0, 22842, 0, 8721, 0, 1152, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 60*x^8 + 1152*x^6 + 8721*x^4 + 22842*x^2 + 10449)
 
gp: K = bnfinit(x^10 + 60*x^8 + 1152*x^6 + 8721*x^4 + 22842*x^2 + 10449, 1)
 

Normalized defining polynomial

\( x^{10} + 60 x^{8} + 1152 x^{6} + 8721 x^{4} + 22842 x^{2} + 10449 \)

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:  \(-4767194945484112896=-\,2^{10}\cdot 3^{5}\cdot 19^{4}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.76$
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:  $2, 3, 19, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{135} a^{6} + \frac{1}{45} a^{4} + \frac{1}{15} a^{2} + \frac{1}{5}$, $\frac{1}{135} a^{7} + \frac{1}{45} a^{5} + \frac{1}{15} a^{3} + \frac{1}{5} a$, $\frac{1}{7695} a^{8} - \frac{8}{2565} a^{6} - \frac{28}{855} a^{4} - \frac{11}{95} a^{2} - \frac{29}{95}$, $\frac{1}{7695} a^{9} - \frac{8}{2565} a^{7} - \frac{28}{855} a^{5} - \frac{11}{95} a^{3} - \frac{29}{95} a$

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

Class group and class number

$C_{2}\times C_{534}$, which has order $1068$ (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:  $4$
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:  \( \frac{2}{7695} a^{8} + \frac{22}{2565} a^{6} - \frac{2}{95} a^{4} - \frac{313}{285} a^{2} - \frac{23}{19} \),  \( \frac{2}{7695} a^{8} + \frac{22}{2565} a^{6} - \frac{2}{95} a^{4} - \frac{313}{285} a^{2} - \frac{42}{19} \),  \( \frac{1}{7695} a^{8} + \frac{2}{171} a^{6} + \frac{59}{171} a^{4} + \frac{70}{19} a^{2} + \frac{1054}{95} \),  \( \frac{16}{7695} a^{8} + \frac{28}{285} a^{6} + \frac{977}{855} a^{4} + \frac{992}{285} a^{2} + \frac{106}{95} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 269.6129204212416 \) (assuming GRH)
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

\(\Q(\sqrt{-129}) \), 5.5.667489.1

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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$