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Properties

Label 10.0.70836297650...0963.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,11^{9}\cdot 313^{5}$
Root discriminant $153.12$
Ramified primes $11, 313$
Class number $118096$ (GRH)
Class group $[118096]$ (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![33831555187, -2072637139, 2072637139, -36809059, 36809059, -268555, 268555, -859, 859, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 859*x^8 - 859*x^7 + 268555*x^6 - 268555*x^5 + 36809059*x^4 - 36809059*x^3 + 2072637139*x^2 - 2072637139*x + 33831555187)
 
gp: K = bnfinit(x^10 - x^9 + 859*x^8 - 859*x^7 + 268555*x^6 - 268555*x^5 + 36809059*x^4 - 36809059*x^3 + 2072637139*x^2 - 2072637139*x + 33831555187, 1)
 

Normalized defining polynomial

\( x^{10} - x^{9} + 859 x^{8} - 859 x^{7} + 268555 x^{6} - 268555 x^{5} + 36809059 x^{4} - 36809059 x^{3} + 2072637139 x^{2} - 2072637139 x + 33831555187 \)

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:  \(-7083629765056720310963=-\,11^{9}\cdot 313^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.12$
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:  $11, 313$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3443=11\cdot 313\)
Dirichlet character group:    $\lbrace$$\chi_{3443}(1,·)$, $\chi_{3443}(1251,·)$, $\chi_{3443}(2503,·)$, $\chi_{3443}(940,·)$, $\chi_{3443}(2192,·)$, $\chi_{3443}(3442,·)$, $\chi_{3443}(1877,·)$, $\chi_{3443}(1879,·)$, $\chi_{3443}(1564,·)$, $\chi_{3443}(1566,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3459180583} a^{6} - \frac{220612406}{3459180583} a^{5} + \frac{468}{3459180583} a^{4} + \frac{440676235}{3459180583} a^{3} + \frac{54756}{3459180583} a^{2} - \frac{219059500}{3459180583} a + \frac{949104}{3459180583}$, $\frac{1}{3459180583} a^{7} + \frac{546}{3459180583} a^{5} - \frac{88135247}{3459180583} a^{4} + \frac{85176}{3459180583} a^{3} + \frac{175247600}{3459180583} a^{2} + \frac{3321864}{3459180583} a - \frac{83704766}{3459180583}$, $\frac{1}{3459180583} a^{8} - \frac{705081976}{3459180583} a^{5} - \frac{170352}{3459180583} a^{4} + \frac{1708664100}{3459180583} a^{3} - \frac{26574912}{3459180583} a^{2} - \frac{1548538171}{3459180583} a - \frac{518210784}{3459180583}$, $\frac{1}{3459180583} a^{9} - \frac{219024}{3459180583} a^{5} - \frac{394307100}{3459180583} a^{4} - \frac{45556992}{3459180583} a^{3} + \frac{1464833405}{3459180583} a^{2} + \frac{1460367559}{3459180583} a + \frac{344065239}{3459180583}$

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

Class group and class number

$C_{118096}$, which has order $118096$ (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

\(\Q(\sqrt{-3443}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
313Data not computed