↑

Properties

Label 15.7.87939541620...7424.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{17}\cdot 3^{15}\cdot 881^{6}$
Root discriminant $99.15$
Ramified primes $2, 3, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T90

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![384, -1248, -504, 2036, 174, -1863, 160, 945, -108, -330, 24, 90, -2, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 - 2*x^12 + 90*x^11 + 24*x^10 - 330*x^9 - 108*x^8 + 945*x^7 + 160*x^6 - 1863*x^5 + 174*x^4 + 2036*x^3 - 504*x^2 - 1248*x + 384)
 
gp: K = bnfinit(x^15 - 15*x^13 - 2*x^12 + 90*x^11 + 24*x^10 - 330*x^9 - 108*x^8 + 945*x^7 + 160*x^6 - 1863*x^5 + 174*x^4 + 2036*x^3 - 504*x^2 - 1248*x + 384, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} - 2 x^{12} + 90 x^{11} + 24 x^{10} - 330 x^{9} - 108 x^{8} + 945 x^{7} + 160 x^{6} - 1863 x^{5} + 174 x^{4} + 2036 x^{3} - 504 x^{2} - 1248 x + 384 \)

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:  \(879395416202630789696044007424=2^{17}\cdot 3^{15}\cdot 881^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.15$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{4}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} + \frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{3}$, $\frac{1}{64} a^{12} + \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{13}{64} a^{4} - \frac{1}{4} a^{3} + \frac{5}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{32} a^{9} + \frac{1}{64} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} + \frac{1}{64} a^{10} - \frac{1}{64} a^{9} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{3}{128} a^{6} + \frac{3}{128} a^{5} - \frac{7}{32} a^{4} + \frac{7}{32} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$

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:  $10$
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:  \( 21229954529.5 \) (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 $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.6.11.12$x^{6} + 2 x^{2} + 2$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
881Data not computed