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Properties

Label 15.15.1584426204...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{15}\cdot 13^{4}\cdot 37^{5}$
Root discriminant $75.86$
Ramified primes $2, 5, 13, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T49

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-676, -6760, -8450, 25285, 20150, -38171, -10120, 22000, 1710, -5825, -82, 750, 0, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 45*x^13 + 750*x^11 - 82*x^10 - 5825*x^9 + 1710*x^8 + 22000*x^7 - 10120*x^6 - 38171*x^5 + 20150*x^4 + 25285*x^3 - 8450*x^2 - 6760*x - 676)
 
gp: K = bnfinit(x^15 - 45*x^13 + 750*x^11 - 82*x^10 - 5825*x^9 + 1710*x^8 + 22000*x^7 - 10120*x^6 - 38171*x^5 + 20150*x^4 + 25285*x^3 - 8450*x^2 - 6760*x - 676, 1)
 

Normalized defining polynomial

\( x^{15} - 45 x^{13} + 750 x^{11} - 82 x^{10} - 5825 x^{9} + 1710 x^{8} + 22000 x^{7} - 10120 x^{6} - 38171 x^{5} + 20150 x^{4} + 25285 x^{3} - 8450 x^{2} - 6760 x - 676 \)

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:  \(15844262047016000000000000000=2^{18}\cdot 5^{15}\cdot 13^{4}\cdot 37^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.86$
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:  $2, 5, 13, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} - \frac{6}{13} a^{11} - \frac{4}{13} a^{9} - \frac{4}{13} a^{8} - \frac{1}{13} a^{7} - \frac{6}{13} a^{6} + \frac{4}{13} a^{5} - \frac{6}{13} a^{4} - \frac{3}{13} a^{3}$, $\frac{1}{2328142763005136038} a^{14} + \frac{33549737348062301}{1164071381502568019} a^{13} - \frac{861592888278881999}{2328142763005136038} a^{12} - \frac{418809346496495525}{1164071381502568019} a^{11} + \frac{62316906030793503}{1164071381502568019} a^{10} - \frac{150524991584225879}{1164071381502568019} a^{9} + \frac{1121747957679867319}{2328142763005136038} a^{8} + \frac{465807975279508441}{1164071381502568019} a^{7} - \frac{294622473067007810}{1164071381502568019} a^{6} + \frac{501518282017246291}{1164071381502568019} a^{5} - \frac{255879174697925195}{2328142763005136038} a^{4} + \frac{226383485497879756}{1164071381502568019} a^{3} + \frac{81024451885778669}{179087904846548926} a^{2} + \frac{27106307411683846}{89543952423274463} a - \frac{5209001312256731}{89543952423274463}$

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:  \( 5493792487.66 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T49:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 29 conjugacy class representatives for [5^3:4]S(3)
Character table for [5^3:4]S(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$5$5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.12$x^{10} + 20 x^{6} + 10 x^{5} + 25 x^{2} + 100 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
37Data not computed