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Properties

Label 15.3.12936093166...4496.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{15}\cdot 3^{15}\cdot 257^{2}\cdot 1609^{3}$
Root discriminant $55.05$
Ramified primes $2, 3, 257, 1609$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T83

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8224, 0, 55512, -992, -96852, -54639, 38392, 31182, -5940, -6915, 366, 756, -6, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 42*x^13 - 6*x^12 + 756*x^11 + 366*x^10 - 6915*x^9 - 5940*x^8 + 31182*x^7 + 38392*x^6 - 54639*x^5 - 96852*x^4 - 992*x^3 + 55512*x^2 - 8224)
 
gp: K = bnfinit(x^15 - 42*x^13 - 6*x^12 + 756*x^11 + 366*x^10 - 6915*x^9 - 5940*x^8 + 31182*x^7 + 38392*x^6 - 54639*x^5 - 96852*x^4 - 992*x^3 + 55512*x^2 - 8224, 1)
 

Normalized defining polynomial

\( x^{15} - 42 x^{13} - 6 x^{12} + 756 x^{11} + 366 x^{10} - 6915 x^{9} - 5940 x^{8} + 31182 x^{7} + 38392 x^{6} - 54639 x^{5} - 96852 x^{4} - 992 x^{3} + 55512 x^{2} - 8224 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(129360931662155595885674496=2^{15}\cdot 3^{15}\cdot 257^{2}\cdot 1609^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.05$
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:  $2, 3, 257, 1609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{38899924619127666619224592} a^{14} + \frac{73242462544489468256721}{9724981154781916654806148} a^{13} - \frac{1518964666040288883994085}{19449962309563833309612296} a^{12} - \frac{4859178610258404410539067}{19449962309563833309612296} a^{11} + \frac{2618644400566671664141911}{9724981154781916654806148} a^{10} + \frac{2232026602531849479297999}{19449962309563833309612296} a^{9} + \frac{17613038174068361479650805}{38899924619127666619224592} a^{8} - \frac{967142948448383238167515}{2431245288695479163701537} a^{7} - \frac{1078929980318668313507841}{19449962309563833309612296} a^{6} - \frac{1414368090482389534381735}{4862490577390958327403074} a^{5} - \frac{7042511018019905036499647}{38899924619127666619224592} a^{4} - \frac{826908599551420705777909}{2431245288695479163701537} a^{3} + \frac{111423253173548223301146}{2431245288695479163701537} a^{2} + \frac{414856229151699195855723}{2431245288695479163701537} a - \frac{948890396027498525470731}{2431245288695479163701537}$

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:  $8$
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:  \( 14916767.2024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T83:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 58320
The 72 conjugacy class representatives for [3^5:2]S(5) are not computed
Character table for [3^5:2]S(5) is not computed

Intermediate fields

5.1.1609.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
3Data not computed
257Data not computed
1609Data not computed