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Properties

Label 15.15.9876832533...2441.1
Degree $15$
Signature $[15, 0]$
Discriminant $61^{14}$
Root discriminant $46.38$
Ramified prime $61$
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![-1, 42, -262, 316, 908, -1721, -956, 2307, 592, -1193, -182, 276, 23, -28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 28*x^13 + 23*x^12 + 276*x^11 - 182*x^10 - 1193*x^9 + 592*x^8 + 2307*x^7 - 956*x^6 - 1721*x^5 + 908*x^4 + 316*x^3 - 262*x^2 + 42*x - 1)
 
gp: K = bnfinit(x^15 - x^14 - 28*x^13 + 23*x^12 + 276*x^11 - 182*x^10 - 1193*x^9 + 592*x^8 + 2307*x^7 - 956*x^6 - 1721*x^5 + 908*x^4 + 316*x^3 - 262*x^2 + 42*x - 1, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 28 x^{13} + 23 x^{12} + 276 x^{11} - 182 x^{10} - 1193 x^{9} + 592 x^{8} + 2307 x^{7} - 956 x^{6} - 1721 x^{5} + 908 x^{4} + 316 x^{3} - 262 x^{2} + 42 x - 1 \)

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:  \(9876832533361318095112441=61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.38$
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:  $61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(61\)
Dirichlet character group:    $\lbrace$$\chi_{61}(1,·)$, $\chi_{61}(34,·)$, $\chi_{61}(9,·)$, $\chi_{61}(42,·)$, $\chi_{61}(12,·)$, $\chi_{61}(13,·)$, $\chi_{61}(15,·)$, $\chi_{61}(16,·)$, $\chi_{61}(20,·)$, $\chi_{61}(22,·)$, $\chi_{61}(25,·)$, $\chi_{61}(56,·)$, $\chi_{61}(57,·)$, $\chi_{61}(58,·)$, $\chi_{61}(47,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{4}{11} a^{8} + \frac{5}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{5}{11} a^{2} - \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{10} - \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{72479} a^{14} + \frac{2137}{72479} a^{13} + \frac{2701}{72479} a^{12} + \frac{2797}{72479} a^{11} - \frac{2550}{72479} a^{10} - \frac{2979}{72479} a^{9} - \frac{31677}{72479} a^{8} - \frac{16270}{72479} a^{7} + \frac{26734}{72479} a^{6} - \frac{36184}{72479} a^{5} - \frac{1664}{72479} a^{4} + \frac{34281}{72479} a^{3} + \frac{3647}{72479} a^{2} - \frac{4352}{72479} a + \frac{18901}{72479}$

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:  \( 87521268.0527 \) (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

3.3.3721.1, 5.5.13845841.1

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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
61Data not computed