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Properties

Label 10.0.26080035472...2144.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,2^{15}\cdot 11^{8}\cdot 13^{5}$
Root discriminant $69.44$
Ramified primes $2, 11, 13$
Class number $1986$ (GRH)
Class group $[1986]$ (GRH)
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![16510339, -1100690, 2652783, -145054, 180115, -7676, 6464, -194, 123, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 123*x^8 - 194*x^7 + 6464*x^6 - 7676*x^5 + 180115*x^4 - 145054*x^3 + 2652783*x^2 - 1100690*x + 16510339)
 
gp: K = bnfinit(x^10 - 2*x^9 + 123*x^8 - 194*x^7 + 6464*x^6 - 7676*x^5 + 180115*x^4 - 145054*x^3 + 2652783*x^2 - 1100690*x + 16510339, 1)
 

Normalized defining polynomial

\( x^{10} - 2 x^{9} + 123 x^{8} - 194 x^{7} + 6464 x^{6} - 7676 x^{5} + 180115 x^{4} - 145054 x^{3} + 2652783 x^{2} - 1100690 x + 16510339 \)

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:  \(-2608003547238662144=-\,2^{15}\cdot 11^{8}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.44$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1144=2^{3}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1144}(1,·)$, $\chi_{1144}(675,·)$, $\chi_{1144}(521,·)$, $\chi_{1144}(779,·)$, $\chi_{1144}(625,·)$, $\chi_{1144}(467,·)$, $\chi_{1144}(155,·)$, $\chi_{1144}(729,·)$, $\chi_{1144}(313,·)$, $\chi_{1144}(883,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{43} a^{8} - \frac{20}{43} a^{7} - \frac{11}{43} a^{6} - \frac{6}{43} a^{5} + \frac{9}{43} a^{4} - \frac{15}{43} a^{3} - \frac{17}{43} a^{2} + \frac{4}{43} a + \frac{11}{43}$, $\frac{1}{5175550975688627291} a^{9} - \frac{33032105159131743}{5175550975688627291} a^{8} + \frac{34847902381863123}{5175550975688627291} a^{7} + \frac{1936661001413473645}{5175550975688627291} a^{6} - \frac{351900121993008274}{5175550975688627291} a^{5} - \frac{1673018212600315725}{5175550975688627291} a^{4} - \frac{403649297594185429}{5175550975688627291} a^{3} + \frac{609428222705075895}{5175550975688627291} a^{2} - \frac{269226566016585688}{5175550975688627291} a - \frac{1444909904305642506}{5175550975688627291}$

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

Class group and class number

$C_{1986}$, which has order $1986$ (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

\(\Q(\sqrt{-26}) \), \(\Q(\zeta_{11})^+\)

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R R ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$13$13.10.5.2$x^{10} - 57122 x^{2} + 2227758$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$