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Properties

Label 10.0.8176563434619904.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{10}\cdot 41^{8}$
Root discriminant $39.02$
Ramified primes $2, 41$
Class number $5$
Class group $[5]$
Galois group $C_{10}$ (as 10T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, 531, 0, 679, 0, 288, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 33*x^8 + 288*x^6 + 679*x^4 + 531*x^2 + 81)
 
gp: K = bnfinit(x^10 + 33*x^8 + 288*x^6 + 679*x^4 + 531*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{10} + 33 x^{8} + 288 x^{6} + 679 x^{4} + 531 x^{2} + 81 \)

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:  \(-8176563434619904=-\,2^{10}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.02$
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:  $2, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(164=2^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{164}(1,·)$, $\chi_{164}(83,·)$, $\chi_{164}(37,·)$, $\chi_{164}(139,·)$, $\chi_{164}(141,·)$, $\chi_{164}(51,·)$, $\chi_{164}(119,·)$, $\chi_{164}(57,·)$, $\chi_{164}(59,·)$, $\chi_{164}(133,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{27} a^{6} + \frac{2}{27} a^{4} + \frac{1}{27} a^{2} + \frac{1}{3}$, $\frac{1}{27} a^{7} + \frac{2}{27} a^{5} + \frac{1}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{9} a^{4} + \frac{7}{27} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{5} + \frac{7}{81} a^{3} - \frac{2}{9} a$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( \frac{5}{27} a^{9} + \frac{158}{27} a^{7} + \frac{1219}{27} a^{5} + \frac{1696}{27} a^{3} + 13 a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{41}{81} a^{9} + \frac{431}{27} a^{7} + \frac{3305}{27} a^{5} + \frac{13298}{81} a^{3} + \frac{218}{9} a \),  \( \frac{5}{27} a^{8} + \frac{158}{27} a^{6} + \frac{1219}{27} a^{4} + \frac{1705}{27} a^{2} + \frac{43}{3} \),  \( \frac{1}{27} a^{8} + \frac{32}{27} a^{6} + \frac{250}{27} a^{4} + \frac{121}{9} a^{2} + 3 \),  \( \frac{17}{27} a^{9} + \frac{536}{27} a^{7} + \frac{4108}{27} a^{5} + \frac{5506}{27} a^{3} + \frac{104}{3} a \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1973.27972308 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}$ (as 10T1):

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

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.2825761.1

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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$41$41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$