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Properties

Label 15.15.1930313361...4928.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{24}\cdot 3^{20}\cdot 53^{9}$
Root discriminant $142.03$
Ramified primes $2, 3, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T85

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36864, 285696, 663552, 63232, -1530432, -1285488, 579072, 717312, -49536, -97160, 1584, 5292, -20, -123, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 123*x^13 - 20*x^12 + 5292*x^11 + 1584*x^10 - 97160*x^9 - 49536*x^8 + 717312*x^7 + 579072*x^6 - 1285488*x^5 - 1530432*x^4 + 63232*x^3 + 663552*x^2 + 285696*x + 36864)
 
gp: K = bnfinit(x^15 - 123*x^13 - 20*x^12 + 5292*x^11 + 1584*x^10 - 97160*x^9 - 49536*x^8 + 717312*x^7 + 579072*x^6 - 1285488*x^5 - 1530432*x^4 + 63232*x^3 + 663552*x^2 + 285696*x + 36864, 1)
 

Normalized defining polynomial

\( x^{15} - 123 x^{13} - 20 x^{12} + 5292 x^{11} + 1584 x^{10} - 97160 x^{9} - 49536 x^{8} + 717312 x^{7} + 579072 x^{6} - 1285488 x^{5} - 1530432 x^{4} + 63232 x^{3} + 663552 x^{2} + 285696 x + 36864 \)

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:  \(193031336102078228638557618044928=2^{24}\cdot 3^{20}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.03$
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:  $2, 3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{9} + \frac{1}{32} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{12} + \frac{1}{128} a^{10} - \frac{1}{32} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{13} - \frac{3}{512} a^{11} - \frac{1}{128} a^{10} + \frac{1}{128} a^{9} + \frac{3}{64} a^{7} - \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{7}{32} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{28675182955952043025913856} a^{14} - \frac{1088392255743203732549}{1194799289831335126079744} a^{13} - \frac{825496900735678035601}{9558394318650681008637952} a^{12} - \frac{35559902513606609860259}{7168795738988010756478464} a^{11} + \frac{26503581319075986338095}{2389598579662670252159488} a^{10} + \frac{14038003109843973369679}{597399644915667563039872} a^{9} + \frac{2813452292367250113107}{3584397869494005378239232} a^{8} - \frac{9198960264916984656703}{149349911228916890759968} a^{7} + \frac{6802528224321319536095}{149349911228916890759968} a^{6} + \frac{179820924440019686839}{37337477807229222689992} a^{5} + \frac{59651773115779852220163}{597399644915667563039872} a^{4} - \frac{6781204686001963273933}{149349911228916890759968} a^{3} + \frac{81787193953082750815199}{224024866843375336139952} a^{2} + \frac{706901846734177171932}{4667184725903652836249} a + \frac{1520813620217238211216}{4667184725903652836249}$

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:  \( 1712775117290 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T85:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 45 conjugacy class representatives for [1/2.S(3)^5]F(5)
Character table for [1/2.S(3)^5]F(5) is not computed

Intermediate fields

5.5.2382032.1

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

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ R $15$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.18.41$x^{8} + 4 x^{6} + 20 x^{4} + 44$$4$$2$$18$$((C_8 : C_2):C_2):C_2$$[2, 3, 3, 7/2]^{4}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.6$x^{12} + 120 x^{11} - 117 x^{10} - 57 x^{9} + 36 x^{8} + 54 x^{7} - 18 x^{6} + 81 x^{5} + 81$$3$$4$$16$12T46$[2, 2]^{8}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.9.2$x^{12} - 106 x^{8} + 2809 x^{4} - 9528128$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$