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Properties

Label 15.11.8244332026...5696.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 3^{16}\cdot 881^{6}$
Root discriminant $84.67$
Ramified primes $2, 3, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T88

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4096, 7168, 45312, -20160, -91328, 34464, 52520, -15448, -9852, 2139, 128, -225, -20, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 17*x^13 - 20*x^12 - 225*x^11 + 128*x^10 + 2139*x^9 - 9852*x^8 - 15448*x^7 + 52520*x^6 + 34464*x^5 - 91328*x^4 - 20160*x^3 + 45312*x^2 + 7168*x - 4096)
 
gp: K = bnfinit(x^15 - 17*x^13 - 20*x^12 - 225*x^11 + 128*x^10 + 2139*x^9 - 9852*x^8 - 15448*x^7 + 52520*x^6 + 34464*x^5 - 91328*x^4 - 20160*x^3 + 45312*x^2 + 7168*x - 4096, 1)
 

Normalized defining polynomial

\( x^{15} - 17 x^{13} - 20 x^{12} - 225 x^{11} + 128 x^{10} + 2139 x^{9} - 9852 x^{8} - 15448 x^{7} + 52520 x^{6} + 34464 x^{5} - 91328 x^{4} - 20160 x^{3} + 45312 x^{2} + 7168 x - 4096 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(82443320268996636534004125696=2^{12}\cdot 3^{16}\cdot 881^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{24} a^{9} - \frac{5}{24} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} - \frac{1}{3} a^{4} + \frac{7}{24} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{48} a^{10} - \frac{5}{48} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} - \frac{1}{6} a^{5} - \frac{17}{48} a^{4} + \frac{1}{4} a^{3} - \frac{1}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{96} a^{11} - \frac{1}{96} a^{9} - \frac{1}{8} a^{8} - \frac{17}{96} a^{7} - \frac{1}{12} a^{6} + \frac{43}{96} a^{5} + \frac{7}{24} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{192} a^{12} - \frac{1}{192} a^{10} - \frac{1}{48} a^{9} - \frac{17}{192} a^{8} - \frac{1}{4} a^{7} + \frac{43}{192} a^{6} - \frac{11}{48} a^{5} + \frac{7}{24} a^{4} - \frac{3}{8} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{1536} a^{13} - \frac{1}{1536} a^{11} + \frac{1}{128} a^{10} + \frac{5}{512} a^{9} + \frac{5}{48} a^{8} - \frac{103}{512} a^{7} + \frac{121}{384} a^{6} + \frac{25}{64} a^{5} - \frac{29}{64} a^{4} + \frac{5}{48} a^{3} - \frac{11}{24} a^{2} - \frac{11}{24} a - \frac{1}{6}$, $\frac{1}{440182010358362185728} a^{14} + \frac{19933615400402449}{110045502589590546432} a^{13} + \frac{69057299033379071}{440182010358362185728} a^{12} - \frac{8515844683302553}{18340917098265091072} a^{11} - \frac{3829226714004178753}{440182010358362185728} a^{10} - \frac{588040446186440507}{36681834196530182144} a^{9} - \frac{4061696603514140903}{146727336786120728576} a^{8} - \frac{1720756441119682249}{27511375647397636608} a^{7} - \frac{8846285833207280051}{55022751294795273216} a^{6} - \frac{5835863704438591501}{18340917098265091072} a^{5} + \frac{2764258777616673295}{6877843911849409152} a^{4} + \frac{854874363833333513}{6877843911849409152} a^{3} - \frac{3189986625845186231}{6877843911849409152} a^{2} + \frac{369000527913276941}{859730488981176144} a + \frac{37745420073095299}{143288414830196024}$

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:  $12$
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:  \( 9860259684.86 \) (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

5.5.3104644.1

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.10.1$x^{6} + 2 x^{5} + 2 x^{4} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
881Data not computed