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Properties

Label 15.7.10962798110...9024.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{8}\cdot 3^{14}\cdot 11^{6}\cdot 131^{6}$
Root discriminant $74.02$
Ramified primes $2, 3, 11, 131$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T88

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-864, -6960, 3312, 5992, -2694, -3915, 1016, 1629, -324, -406, 72, 90, -6, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 - 6*x^12 + 90*x^11 + 72*x^10 - 406*x^9 - 324*x^8 + 1629*x^7 + 1016*x^6 - 3915*x^5 - 2694*x^4 + 5992*x^3 + 3312*x^2 - 6960*x - 864)
 
gp: K = bnfinit(x^15 - 15*x^13 - 6*x^12 + 90*x^11 + 72*x^10 - 406*x^9 - 324*x^8 + 1629*x^7 + 1016*x^6 - 3915*x^5 - 2694*x^4 + 5992*x^3 + 3312*x^2 - 6960*x - 864, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} - 6 x^{12} + 90 x^{11} + 72 x^{10} - 406 x^{9} - 324 x^{8} + 1629 x^{7} + 1016 x^{6} - 3915 x^{5} - 2694 x^{4} + 5992 x^{3} + 3312 x^{2} - 6960 x - 864 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10962798110524462576263969024=2^{8}\cdot 3^{14}\cdot 11^{6}\cdot 131^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.02$
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:  $2, 3, 11, 131$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{8} + \frac{1}{64} a^{7} + \frac{1}{32} a^{5} + \frac{5}{64} a^{4} + \frac{1}{64} a^{3} + \frac{1}{32} a^{2} + \frac{3}{16} a + \frac{1}{8}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{3}{64} a^{5} + \frac{1}{16} a^{4} + \frac{1}{64} a^{3} - \frac{5}{32} a^{2} - \frac{1}{16} a + \frac{1}{8}$, $\frac{1}{128} a^{10} - \frac{1}{128} a^{8} + \frac{1}{64} a^{7} + \frac{3}{128} a^{6} - \frac{1}{32} a^{5} + \frac{1}{128} a^{4} + \frac{5}{64} a^{3} - \frac{1}{32} a^{2} - \frac{1}{16} a$, $\frac{1}{256} a^{11} - \frac{1}{256} a^{10} + \frac{1}{256} a^{9} - \frac{1}{256} a^{8} - \frac{5}{256} a^{7} - \frac{3}{256} a^{6} + \frac{3}{256} a^{5} - \frac{19}{256} a^{4} - \frac{1}{16} a^{3} + \frac{1}{32} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{512} a^{12} - \frac{1}{128} a^{9} + \frac{1}{256} a^{8} - \frac{3}{128} a^{7} + \frac{1}{64} a^{6} + \frac{5}{128} a^{5} - \frac{11}{512} a^{4} + \frac{11}{128} a^{3} + \frac{1}{32} a^{2} + \frac{13}{32} a + \frac{15}{32}$, $\frac{1}{1024} a^{13} + \frac{1}{512} a^{9} - \frac{1}{128} a^{8} + \frac{3}{128} a^{7} - \frac{1}{32} a^{6} - \frac{11}{1024} a^{5} + \frac{11}{128} a^{4} + \frac{1}{16} a^{3} - \frac{15}{64} a^{2} + \frac{3}{64} a + \frac{5}{16}$, $\frac{1}{6144} a^{14} - \frac{1}{2048} a^{13} - \frac{1}{1024} a^{12} - \frac{1}{1024} a^{10} + \frac{7}{1024} a^{9} - \frac{1}{1536} a^{8} - \frac{17}{2048} a^{6} - \frac{223}{6144} a^{5} + \frac{43}{1024} a^{4} - \frac{17}{256} a^{3} - \frac{7}{192} a^{2} - \frac{9}{128} a + \frac{11}{64}$

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:  $10$
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:  \( 2171299929.11 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T88:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 233280
The 48 conjugacy class representatives for [1/2.S(3)^5]A(5)
Character table for [1/2.S(3)^5]A(5) is not computed

Intermediate fields

5.5.8305924.1

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 $15$ $15$ R $15$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.9.9.4$x^{9} + 3 x^{6} + 9 x^{4} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
131Data not computed