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Properties

Label 15.11.1109203183...3524.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{2}\cdot 3^{20}\cdot 17^{2}\cdot 31^{3}\cdot 9739^{3}$
Root discriminant $86.36$
Ramified primes $2, 3, 17, 31, 9739$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T91

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![544, 816, -17136, 28312, -35190, 28314, 41392, -31521, -10512, 5806, 1248, -90, -64, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 33*x^13 - 64*x^12 - 90*x^11 + 1248*x^10 + 5806*x^9 - 10512*x^8 - 31521*x^7 + 41392*x^6 + 28314*x^5 - 35190*x^4 + 28312*x^3 - 17136*x^2 + 816*x + 544)
 
gp: K = bnfinit(x^15 - 33*x^13 - 64*x^12 - 90*x^11 + 1248*x^10 + 5806*x^9 - 10512*x^8 - 31521*x^7 + 41392*x^6 + 28314*x^5 - 35190*x^4 + 28312*x^3 - 17136*x^2 + 816*x + 544, 1)
 

Normalized defining polynomial

\( x^{15} - 33 x^{13} - 64 x^{12} - 90 x^{11} + 1248 x^{10} + 5806 x^{9} - 10512 x^{8} - 31521 x^{7} + 41392 x^{6} + 28314 x^{5} - 35190 x^{4} + 28312 x^{3} - 17136 x^{2} + 816 x + 544 \)

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:  \(110920318379317587571477873524=2^{2}\cdot 3^{20}\cdot 17^{2}\cdot 31^{3}\cdot 9739^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.36$
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:  $2, 3, 17, 31, 9739$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{22846637778539423463353286772496} a^{14} - \frac{206282052070292019248039197273}{5711659444634855865838321693124} a^{13} + \frac{1460425918441431045699317284743}{22846637778539423463353286772496} a^{12} + \frac{696865516273063468404455232531}{5711659444634855865838321693124} a^{11} + \frac{563262259421857207979545255091}{11423318889269711731676643386248} a^{10} + \frac{187520603149300097926980031385}{1427914861158713966459580423281} a^{9} - \frac{554731821947269536377802785}{11423318889269711731676643386248} a^{8} - \frac{304396187349894849070037541661}{1427914861158713966459580423281} a^{7} - \frac{6910789436834403276656281587785}{22846637778539423463353286772496} a^{6} - \frac{818529922127292750229114522697}{5711659444634855865838321693124} a^{5} + \frac{4346052860228027163132109763181}{11423318889269711731676643386248} a^{4} + \frac{285326677319648379242650907213}{11423318889269711731676643386248} a^{3} - \frac{127658158629878894077864498385}{1427914861158713966459580423281} a^{2} + \frac{252691128755742846233495690345}{1427914861158713966459580423281} a + \frac{373844836513799491243762490059}{1427914861158713966459580423281}$

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:  \( 4145512912.37 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T91:

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 [1/2.S(3)^5]S(5) are not computed
Character table for [1/2.S(3)^5]S(5) is not computed

Intermediate fields

5.5.301909.1

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 $15$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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$$[\ ]_{3}^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
9739Data not computed