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Properties

Label 15.15.2262214114...1609.1
Degree $15$
Signature $[15, 0]$
Discriminant $13^{10}\cdot 71^{12}$
Root discriminant $167.35$
Ramified primes $13, 71$
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![261871, 749403, -853079, -1587370, 1390892, 806127, -857980, -60296, 198291, -27588, -14720, 3348, 334, -105, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 105*x^13 + 334*x^12 + 3348*x^11 - 14720*x^10 - 27588*x^9 + 198291*x^8 - 60296*x^7 - 857980*x^6 + 806127*x^5 + 1390892*x^4 - 1587370*x^3 - 853079*x^2 + 749403*x + 261871)
 
gp: K = bnfinit(x^15 - 2*x^14 - 105*x^13 + 334*x^12 + 3348*x^11 - 14720*x^10 - 27588*x^9 + 198291*x^8 - 60296*x^7 - 857980*x^6 + 806127*x^5 + 1390892*x^4 - 1587370*x^3 - 853079*x^2 + 749403*x + 261871, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 105 x^{13} + 334 x^{12} + 3348 x^{11} - 14720 x^{10} - 27588 x^{9} + 198291 x^{8} - 60296 x^{7} - 857980 x^{6} + 806127 x^{5} + 1390892 x^{4} - 1587370 x^{3} - 853079 x^{2} + 749403 x + 261871 \)

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:  \(2262214114345307242786511343301609=13^{10}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.35$
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:  $13, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(923=13\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{923}(640,·)$, $\chi_{923}(1,·)$, $\chi_{923}(835,·)$, $\chi_{923}(196,·)$, $\chi_{923}(711,·)$, $\chi_{923}(360,·)$, $\chi_{923}(412,·)$, $\chi_{923}(906,·)$, $\chi_{923}(289,·)$, $\chi_{923}(625,·)$, $\chi_{923}(451,·)$, $\chi_{923}(341,·)$, $\chi_{923}(664,·)$, $\chi_{923}(380,·)$, $\chi_{923}(573,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{23} a^{12} - \frac{2}{23} a^{11} - \frac{1}{23} a^{10} + \frac{10}{23} a^{9} + \frac{3}{23} a^{8} + \frac{6}{23} a^{7} - \frac{8}{23} a^{6} + \frac{8}{23} a^{5} + \frac{1}{23} a^{3} - \frac{8}{23} a^{2} + \frac{2}{23} a - \frac{7}{23}$, $\frac{1}{23} a^{13} - \frac{5}{23} a^{11} + \frac{8}{23} a^{10} - \frac{11}{23} a^{8} + \frac{4}{23} a^{7} - \frac{8}{23} a^{6} - \frac{7}{23} a^{5} + \frac{1}{23} a^{4} - \frac{6}{23} a^{3} + \frac{9}{23} a^{2} - \frac{3}{23} a + \frac{9}{23}$, $\frac{1}{317454597905207700300547750145941} a^{14} + \frac{1290880280068427372073341482233}{317454597905207700300547750145941} a^{13} + \frac{3211508902673159521597946964608}{317454597905207700300547750145941} a^{12} - \frac{79875508491409509489203780821744}{317454597905207700300547750145941} a^{11} - \frac{97199526350326365414433396630242}{317454597905207700300547750145941} a^{10} + \frac{125260240718843398357213588738988}{317454597905207700300547750145941} a^{9} + \frac{106744168370214074488306793605243}{317454597905207700300547750145941} a^{8} - \frac{31586720878744502956075381250931}{317454597905207700300547750145941} a^{7} - \frac{142797281370666712422644870424017}{317454597905207700300547750145941} a^{6} - \frac{39208318452717945309173236643536}{317454597905207700300547750145941} a^{5} + \frac{139821539750802904365169360626771}{317454597905207700300547750145941} a^{4} + \frac{48105510983560787548627523691326}{317454597905207700300547750145941} a^{3} + \frac{76495363702660824519671725412815}{317454597905207700300547750145941} a^{2} + \frac{42063749507244493622089124535285}{317454597905207700300547750145941} a - \frac{29606685315315093723721659300508}{317454597905207700300547750145941}$

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:  \( 444156020841.87354 \) (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, 5.5.25411681.1

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 $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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}$$[\ ]_{3}^{5}$
$71$71.15.12.1$x^{15} + 710 x^{10} + 95779 x^{5} + 11453152$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$