↑

Properties

Label 10.0.66319952941...3619.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,11^{8}\cdot 499^{5}$
Root discriminant $152.11$
Ramified primes $11, 499$
Class number $134013$ (GRH)
Class group $[134013]$ (GRH)
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32508046999, -744124505, 1255906627, -23314473, 19624244, -277040, 155010, -1480, 619, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 619*x^8 - 1480*x^7 + 155010*x^6 - 277040*x^5 + 19624244*x^4 - 23314473*x^3 + 1255906627*x^2 - 744124505*x + 32508046999)
 
gp: K = bnfinit(x^10 - 3*x^9 + 619*x^8 - 1480*x^7 + 155010*x^6 - 277040*x^5 + 19624244*x^4 - 23314473*x^3 + 1255906627*x^2 - 744124505*x + 32508046999, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} + 619 x^{8} - 1480 x^{7} + 155010 x^{6} - 277040 x^{5} + 19624244 x^{4} - 23314473 x^{3} + 1255906627 x^{2} - 744124505 x + 32508046999 \)

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:  \(-6631995294177230343619=-\,11^{8}\cdot 499^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.11$
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:  $11, 499$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5489=11\cdot 499\)
Dirichlet character group:    $\lbrace$$\chi_{5489}(1,·)$, $\chi_{5489}(3492,·)$, $\chi_{5489}(999,·)$, $\chi_{5489}(1995,·)$, $\chi_{5489}(4492,·)$, $\chi_{5489}(2993,·)$, $\chi_{5489}(498,·)$, $\chi_{5489}(2995,·)$, $\chi_{5489}(500,·)$, $\chi_{5489}(3991,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1044713030095289624831481523} a^{9} - \frac{267990525188133702078761398}{1044713030095289624831481523} a^{8} + \frac{236245992652785535635834420}{1044713030095289624831481523} a^{7} + \frac{154777127006212200591563913}{1044713030095289624831481523} a^{6} + \frac{344677031713474707736601880}{1044713030095289624831481523} a^{5} + \frac{102934125475721249690694794}{1044713030095289624831481523} a^{4} - \frac{244178627005286660781589813}{1044713030095289624831481523} a^{3} - \frac{514855977764222901236966787}{1044713030095289624831481523} a^{2} - \frac{395853981814823259150181007}{1044713030095289624831481523} a - \frac{254872611797368394907041968}{1044713030095289624831481523}$

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

Class group and class number

$C_{134013}$, which has order $134013$ (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:  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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26.1711060094 \) (assuming GRH)
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

\(\Q(\sqrt{-499}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
499Data not computed