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Properties

Label 15.13.1717569172...5952.1
Degree $15$
Signature $[13, 1]$
Discriminant $-\,2^{8}\cdot 3^{15}\cdot 881^{6}$
Root discriminant $65.41$
Ramified primes $2, 3, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T90

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1152, -8640, -25920, -36384, -14184, 24948, 32288, 6300, -9504, -4436, 948, 711, -30, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 45*x^13 - 30*x^12 + 711*x^11 + 948*x^10 - 4436*x^9 - 9504*x^8 + 6300*x^7 + 32288*x^6 + 24948*x^5 - 14184*x^4 - 36384*x^3 - 25920*x^2 - 8640*x - 1152)
 
gp: K = bnfinit(x^15 - 45*x^13 - 30*x^12 + 711*x^11 + 948*x^10 - 4436*x^9 - 9504*x^8 + 6300*x^7 + 32288*x^6 + 24948*x^5 - 14184*x^4 - 36384*x^3 - 25920*x^2 - 8640*x - 1152, 1)
 

Normalized defining polynomial

\( x^{15} - 45 x^{13} - 30 x^{12} + 711 x^{11} + 948 x^{10} - 4436 x^{9} - 9504 x^{8} + 6300 x^{7} + 32288 x^{6} + 24948 x^{5} - 14184 x^{4} - 36384 x^{3} - 25920 x^{2} - 8640 x - 1152 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1717569172270763261125085952=-\,2^{8}\cdot 3^{15}\cdot 881^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.41$
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:  $2, 3, 881$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{12} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} + \frac{5}{24} a^{8} - \frac{5}{48} a^{7} + \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{192} a^{12} - \frac{1}{96} a^{11} - \frac{5}{192} a^{10} - \frac{1}{48} a^{9} - \frac{7}{64} a^{8} - \frac{5}{96} a^{7} + \frac{1}{16} a^{6} + \frac{1}{24} a^{5} - \frac{5}{48} a^{4} - \frac{11}{24} a^{3} - \frac{1}{16} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{1536} a^{13} - \frac{1}{384} a^{12} + \frac{5}{512} a^{11} - \frac{5}{768} a^{10} + \frac{1}{512} a^{9} - \frac{17}{96} a^{8} + \frac{5}{32} a^{7} + \frac{11}{48} a^{6} - \frac{73}{384} a^{5} + \frac{19}{96} a^{4} + \frac{97}{384} a^{3} + \frac{19}{64} a^{2} - \frac{3}{32} a - \frac{7}{16}$, $\frac{1}{12288} a^{14} + \frac{1}{6144} a^{13} + \frac{23}{12288} a^{12} + \frac{1}{768} a^{11} + \frac{167}{12288} a^{10} + \frac{193}{6144} a^{9} + \frac{13}{256} a^{8} + \frac{11}{192} a^{7} + \frac{263}{3072} a^{6} - \frac{53}{1536} a^{5} + \frac{451}{1024} a^{4} + \frac{23}{384} a^{3} + \frac{23}{128} a^{2} + \frac{3}{8} a + \frac{19}{64}$

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:  $13$
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:  \( 4983572147.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T90:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 72 conjugacy class representatives for [S(3)^5]A(5)=S(3)wrA(5) are not computed
Character table for [S(3)^5]A(5)=S(3)wrA(5) is not computed

Intermediate fields

5.5.3104644.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 45 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ $15$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.10.4$x^{6} + 3 x^{3} + 72$$3$$2$$10$$S_3^2$$[3/2, 5/2]_{2}^{2}$
881Data not computed