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Properties

Label 15.15.2224952409...5489.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 31^{12}$
Root discriminant $57.08$
Ramified primes $7, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![211, 2903, 10091, 4468, -22698, -21477, 14370, 18376, -2469, -5782, -86, 796, 44, -47, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 47*x^13 + 44*x^12 + 796*x^11 - 86*x^10 - 5782*x^9 - 2469*x^8 + 18376*x^7 + 14370*x^6 - 21477*x^5 - 22698*x^4 + 4468*x^3 + 10091*x^2 + 2903*x + 211)
 
gp: K = bnfinit(x^15 - 2*x^14 - 47*x^13 + 44*x^12 + 796*x^11 - 86*x^10 - 5782*x^9 - 2469*x^8 + 18376*x^7 + 14370*x^6 - 21477*x^5 - 22698*x^4 + 4468*x^3 + 10091*x^2 + 2903*x + 211, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 47 x^{13} + 44 x^{12} + 796 x^{11} - 86 x^{10} - 5782 x^{9} - 2469 x^{8} + 18376 x^{7} + 14370 x^{6} - 21477 x^{5} - 22698 x^{4} + 4468 x^{3} + 10091 x^{2} + 2903 x + 211 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(222495240978703757087365489=7^{10}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.08$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(217=7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{217}(128,·)$, $\chi_{217}(1,·)$, $\chi_{217}(2,·)$, $\chi_{217}(163,·)$, $\chi_{217}(4,·)$, $\chi_{217}(32,·)$, $\chi_{217}(8,·)$, $\chi_{217}(64,·)$, $\chi_{217}(39,·)$, $\chi_{217}(109,·)$, $\chi_{217}(78,·)$, $\chi_{217}(16,·)$, $\chi_{217}(156,·)$, $\chi_{217}(190,·)$, $\chi_{217}(95,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a$, $\frac{1}{8237701973762799473665} a^{14} - \frac{440152523535910325399}{8237701973762799473665} a^{13} + \frac{272840784304635863434}{8237701973762799473665} a^{12} - \frac{720666254543388573137}{8237701973762799473665} a^{11} + \frac{2626997176095481103972}{8237701973762799473665} a^{10} + \frac{788946092125315780063}{1647540394752559894733} a^{9} + \frac{712769166831074708796}{1647540394752559894733} a^{8} + \frac{2445441931560565528873}{8237701973762799473665} a^{7} + \frac{3770558774543680284258}{8237701973762799473665} a^{6} - \frac{15408438078510230235}{1647540394752559894733} a^{5} + \frac{1747273675282373770352}{8237701973762799473665} a^{4} - \frac{610714609270695421387}{1647540394752559894733} a^{3} + \frac{1868520317107084507432}{8237701973762799473665} a^{2} - \frac{869677519312827512298}{8237701973762799473665} a - \frac{3573798774959395553}{7808248316362843103}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 109278274.626 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.5.923521.1

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 $15$ $15$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ $15$ $15$

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
7Data not computed
31Data not computed