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Properties

Label 15.15.2138179265...4929.2
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 31^{14}$
Root discriminant $90.23$
Ramified primes $7, 31$
Class number $3$ (GRH)
Class group $[3]$ (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![43525, -166230, 38908, 371340, -156827, -330836, 91023, 133041, -17536, -24958, 895, 2093, 13, -76, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 76*x^13 + 13*x^12 + 2093*x^11 + 895*x^10 - 24958*x^9 - 17536*x^8 + 133041*x^7 + 91023*x^6 - 330836*x^5 - 156827*x^4 + 371340*x^3 + 38908*x^2 - 166230*x + 43525)
 
gp: K = bnfinit(x^15 - x^14 - 76*x^13 + 13*x^12 + 2093*x^11 + 895*x^10 - 24958*x^9 - 17536*x^8 + 133041*x^7 + 91023*x^6 - 330836*x^5 - 156827*x^4 + 371340*x^3 + 38908*x^2 - 166230*x + 43525, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 76 x^{13} + 13 x^{12} + 2093 x^{11} + 895 x^{10} - 24958 x^{9} - 17536 x^{8} + 133041 x^{7} + 91023 x^{6} - 330836 x^{5} - 156827 x^{4} + 371340 x^{3} + 38908 x^{2} - 166230 x + 43525 \)

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:  \(213817926580534310560958234929=7^{10}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.23$
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$$\chi_{217}(64,·)$, $\chi_{217}(1,·)$, $\chi_{217}(193,·)$, $\chi_{217}(8,·)$, $\chi_{217}(9,·)$, $\chi_{217}(142,·)$, $\chi_{217}(81,·)$, $\chi_{217}(51,·)$, $\chi_{217}(78,·)$, $\chi_{217}(214,·)$, $\chi_{217}(72,·)$, $\chi_{217}(25,·)$, $\chi_{217}(200,·)$, $\chi_{217}(190,·)$, $\chi_{217}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{8} - \frac{1}{25} a^{7} - \frac{2}{25} a^{6} - \frac{1}{25} a^{5} + \frac{3}{25} a^{4} + \frac{11}{25} a^{3} - \frac{3}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{8} + \frac{1}{25} a^{4}$, $\frac{1}{125} a^{13} + \frac{2}{125} a^{12} - \frac{1}{125} a^{11} - \frac{2}{125} a^{10} - \frac{2}{125} a^{9} + \frac{6}{125} a^{8} - \frac{3}{125} a^{7} - \frac{6}{125} a^{6} + \frac{6}{125} a^{5} + \frac{42}{125} a^{4} + \frac{29}{125} a^{3} + \frac{58}{125} a^{2} - \frac{6}{25} a + \frac{1}{5}$, $\frac{1}{12773566250476412646875} a^{14} + \frac{99766322776928588}{12773566250476412646875} a^{13} + \frac{236059212818151593881}{12773566250476412646875} a^{12} + \frac{4312085617620558047}{12773566250476412646875} a^{11} - \frac{9994182682097851724}{12773566250476412646875} a^{10} + \frac{113352516833469530584}{12773566250476412646875} a^{9} - \frac{201160694284387371732}{12773566250476412646875} a^{8} + \frac{27051220146700215941}{12773566250476412646875} a^{7} + \frac{182503057260153774633}{2554713250095282529375} a^{6} + \frac{439987530672428721208}{12773566250476412646875} a^{5} + \frac{810614065300989624551}{12773566250476412646875} a^{4} + \frac{1450389327849276760462}{12773566250476412646875} a^{3} + \frac{6292396541359504484208}{12773566250476412646875} a^{2} - \frac{417021864076584809216}{2554713250095282529375} a + \frac{77622278859655749406}{510942650019056505875}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 30932433416.2 \) (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

3.3.47089.2, 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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{15}$ R $15$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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