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Properties

Label 15.15.2078285456...0625.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 5^{24}$
Root discriminant $56.82$
Ramified primes $3, 5$
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![-107, -3405, 4575, 15050, -8970, -21975, 6350, 14370, -1965, -4560, 252, 690, -10, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 45*x^13 - 10*x^12 + 690*x^11 + 252*x^10 - 4560*x^9 - 1965*x^8 + 14370*x^7 + 6350*x^6 - 21975*x^5 - 8970*x^4 + 15050*x^3 + 4575*x^2 - 3405*x - 107)
 
gp: K = bnfinit(x^15 - 45*x^13 - 10*x^12 + 690*x^11 + 252*x^10 - 4560*x^9 - 1965*x^8 + 14370*x^7 + 6350*x^6 - 21975*x^5 - 8970*x^4 + 15050*x^3 + 4575*x^2 - 3405*x - 107, 1)
 

Normalized defining polynomial

\( x^{15} - 45 x^{13} - 10 x^{12} + 690 x^{11} + 252 x^{10} - 4560 x^{9} - 1965 x^{8} + 14370 x^{7} + 6350 x^{6} - 21975 x^{5} - 8970 x^{4} + 15050 x^{3} + 4575 x^{2} - 3405 x - 107 \)

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:  \(207828545629978179931640625=3^{20}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.82$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(225=3^{2}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{225}(1,·)$, $\chi_{225}(196,·)$, $\chi_{225}(166,·)$, $\chi_{225}(136,·)$, $\chi_{225}(106,·)$, $\chi_{225}(76,·)$, $\chi_{225}(46,·)$, $\chi_{225}(16,·)$, $\chi_{225}(211,·)$, $\chi_{225}(181,·)$, $\chi_{225}(151,·)$, $\chi_{225}(121,·)$, $\chi_{225}(91,·)$, $\chi_{225}(61,·)$, $\chi_{225}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{4} + \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{965847186647506135343} a^{14} - \frac{56632672193789297996}{965847186647506135343} a^{13} + \frac{9771307762773034581}{965847186647506135343} a^{12} + \frac{228943671714236587051}{965847186647506135343} a^{11} + \frac{338389791224428820969}{965847186647506135343} a^{10} - \frac{369728276168883726672}{965847186647506135343} a^{9} - \frac{354058809460651140793}{965847186647506135343} a^{8} - \frac{122004386134509805840}{965847186647506135343} a^{7} - \frac{370781884756751847849}{965847186647506135343} a^{6} - \frac{42453870685156947338}{137978169521072305049} a^{5} - \frac{342253516537367450516}{965847186647506135343} a^{4} + \frac{320625706093969924182}{965847186647506135343} a^{3} - \frac{422570209016809378918}{965847186647506135343} a^{2} - \frac{37787240247463088960}{137978169521072305049} a - \frac{470760438360246469280}{965847186647506135343}$

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:  \( 111059719.436 \) (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

\(\Q(\zeta_{9})^+\), 5.5.390625.1

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 $15$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
3Data not computed
5Data not computed