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Properties

Label 15.15.3110568743...3769.1
Degree $15$
Signature $[15, 0]$
Discriminant $13^{10}\cdot 41^{12}$
Root discriminant $107.86$
Ramified primes $13, 41$
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![2293, 9, -87985, 33394, 179526, -71155, -119624, 47584, 32905, -13104, -3742, 1536, 154, -69, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 69*x^13 + 154*x^12 + 1536*x^11 - 3742*x^10 - 13104*x^9 + 32905*x^8 + 47584*x^7 - 119624*x^6 - 71155*x^5 + 179526*x^4 + 33394*x^3 - 87985*x^2 + 9*x + 2293)
 
gp: K = bnfinit(x^15 - 2*x^14 - 69*x^13 + 154*x^12 + 1536*x^11 - 3742*x^10 - 13104*x^9 + 32905*x^8 + 47584*x^7 - 119624*x^6 - 71155*x^5 + 179526*x^4 + 33394*x^3 - 87985*x^2 + 9*x + 2293, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 69 x^{13} + 154 x^{12} + 1536 x^{11} - 3742 x^{10} - 13104 x^{9} + 32905 x^{8} + 47584 x^{7} - 119624 x^{6} - 71155 x^{5} + 179526 x^{4} + 33394 x^{3} - 87985 x^{2} + 9 x + 2293 \)

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:  \(3110568743658022425562755753769=13^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.86$
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:  $13, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(533=13\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{533}(256,·)$, $\chi_{533}(1,·)$, $\chi_{533}(100,·)$, $\chi_{533}(133,·)$, $\chi_{533}(92,·)$, $\chi_{533}(42,·)$, $\chi_{533}(139,·)$, $\chi_{533}(365,·)$, $\chi_{533}(16,·)$, $\chi_{533}(529,·)$, $\chi_{533}(469,·)$, $\chi_{533}(406,·)$, $\chi_{533}(508,·)$, $\chi_{533}(510,·)$, $\chi_{533}(165,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{711} a^{13} - \frac{26}{711} a^{12} - \frac{80}{711} a^{11} + \frac{19}{711} a^{10} + \frac{56}{711} a^{9} - \frac{29}{237} a^{8} - \frac{203}{711} a^{7} + \frac{355}{711} a^{6} - \frac{301}{711} a^{5} - \frac{179}{711} a^{4} - \frac{307}{711} a^{3} - \frac{4}{79} a^{2} + \frac{103}{711} a + \frac{106}{711}$, $\frac{1}{344808882295121833872425073} a^{14} - \frac{59344290576217239268618}{344808882295121833872425073} a^{13} + \frac{6212453184224208564109192}{114936294098373944624141691} a^{12} + \frac{22368741827787669200887849}{344808882295121833872425073} a^{11} - \frac{8766741049127772105287996}{114936294098373944624141691} a^{10} + \frac{7743990826637024801784748}{114936294098373944624141691} a^{9} + \frac{78172591727999848887353485}{344808882295121833872425073} a^{8} + \frac{71723469324625492652220431}{344808882295121833872425073} a^{7} + \frac{17757094949454585541196584}{344808882295121833872425073} a^{6} - \frac{97563640207924321482990097}{344808882295121833872425073} a^{5} + \frac{75434169704339790768092464}{344808882295121833872425073} a^{4} - \frac{132744173217091114714415096}{344808882295121833872425073} a^{3} + \frac{48655368268936979562044531}{344808882295121833872425073} a^{2} - \frac{56820351422289270531340567}{114936294098373944624141691} a + \frac{45101900530291364641175245}{114936294098373944624141691}$

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:  \( 25379435677.580284 \) (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

3.3.169.1, 5.5.2825761.1

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 $15$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ R $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
$13$13.15.10.1$x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$41$41.15.12.1$x^{15} + 2665 x^{10} + 1418764 x^{5} + 25589884853$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$