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Properties

Label 10.0.38254197605...9167.1
Degree $10$
Signature $[0, 5]$
Discriminant $-\,3^{5}\cdot 11^{8}\cdot 149^{5}$
Root discriminant $143.97$
Ramified primes $3, 11, 149$
Class number $149870$ (GRH)
Class group $[149870]$ (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![18909114223, -480472389, 812086354, -16760549, 14123866, -222024, 124330, -1324, 554, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 554*x^8 - 1324*x^7 + 124330*x^6 - 222024*x^5 + 14123866*x^4 - 16760549*x^3 + 812086354*x^2 - 480472389*x + 18909114223)
 
gp: K = bnfinit(x^10 - 3*x^9 + 554*x^8 - 1324*x^7 + 124330*x^6 - 222024*x^5 + 14123866*x^4 - 16760549*x^3 + 812086354*x^2 - 480472389*x + 18909114223, 1)
 

Normalized defining polynomial

\( x^{10} - 3 x^{9} + 554 x^{8} - 1324 x^{7} + 124330 x^{6} - 222024 x^{5} + 14123866 x^{4} - 16760549 x^{3} + 812086354 x^{2} - 480472389 x + 18909114223 \)

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:  \(-3825419760558518179167=-\,3^{5}\cdot 11^{8}\cdot 149^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.97$
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:  $3, 11, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4917=3\cdot 11\cdot 149\)
Dirichlet character group:    $\lbrace$$\chi_{4917}(1,·)$, $\chi_{4917}(1340,·)$, $\chi_{4917}(3128,·)$, $\chi_{4917}(4469,·)$, $\chi_{4917}(4471,·)$, $\chi_{4917}(4024,·)$, $\chi_{4917}(2234,·)$, $\chi_{4917}(1787,·)$, $\chi_{4917}(2236,·)$, $\chi_{4917}(895,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{351113571294499826191776343} a^{9} + \frac{79381611977159155032704002}{351113571294499826191776343} a^{8} + \frac{161678439726817613696069091}{351113571294499826191776343} a^{7} + \frac{157396866139341345357168846}{351113571294499826191776343} a^{6} + \frac{105553275226408999877565527}{351113571294499826191776343} a^{5} - \frac{5970040137412840958305112}{351113571294499826191776343} a^{4} - \frac{111720920287203197850485428}{351113571294499826191776343} a^{3} + \frac{85131981437908255122579722}{351113571294499826191776343} a^{2} - \frac{155277684022854601688901917}{351113571294499826191776343} a + \frac{5777744925842917824499285}{15265807447586948964859841}$

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

Class group and class number

$C_{149870}$, which has order $149870$ (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

\(\Q(\sqrt{-447}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$149$149.10.5.2$x^{10} - 492884401 x^{2} + 954717084737$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$