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Properties

Label 15.3.16828429180...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{6}\cdot 5^{19}\cdot 13^{10}$
Root discriminant $56.03$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-40000, 30000, 25000, -15625, 0, 5456, -1220, -1050, 0, 0, -168, 55, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 55*x^11 - 168*x^10 - 1050*x^7 - 1220*x^6 + 5456*x^5 - 15625*x^3 + 25000*x^2 + 30000*x - 40000)
 
gp: K = bnfinit(x^15 + 55*x^11 - 168*x^10 - 1050*x^7 - 1220*x^6 + 5456*x^5 - 15625*x^3 + 25000*x^2 + 30000*x - 40000, 1)
 

Normalized defining polynomial

\( x^{15} + 55 x^{11} - 168 x^{10} - 1050 x^{7} - 1220 x^{6} + 5456 x^{5} - 15625 x^{3} + 25000 x^{2} + 30000 x - 40000 \)

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:  \(168284291807861328125000000=2^{6}\cdot 5^{19}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.03$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{10} a^{7} - \frac{1}{2} a^{5} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{6} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{9}{20} a^{5} + \frac{9}{20} a^{4} + \frac{9}{20} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{8} - \frac{9}{20} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{9}{20} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{9} - \frac{1}{20} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{9}{20} a^{4} - \frac{1}{4} a^{3} + \frac{3}{20} a^{2}$, $\frac{1}{100} a^{13} - \frac{1}{20} a^{9} + \frac{1}{50} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} - \frac{6}{25} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{149391144742561830722000} a^{14} + \frac{6595810395189019897}{1493911447425618307220} a^{13} + \frac{1598551163064112182}{74695572371280915361} a^{12} + \frac{5580843167065430813}{298782289485123661444} a^{11} + \frac{447037126837921589351}{29878228948512366144400} a^{10} + \frac{651043160247964159079}{18673893092820228840250} a^{9} + \frac{6273818298258271}{149391144742561830722} a^{8} + \frac{2468520105712538872}{74695572371280915361} a^{7} - \frac{1280556930106461489251}{2987822894851236614440} a^{6} - \frac{342545828246553994}{1867389309282022884025} a^{5} - \frac{7689649502777497037311}{37347786185640457680500} a^{4} - \frac{53773326357083645588}{373477861856404576805} a^{3} + \frac{2159908013413502729}{11603195708160142192} a^{2} + \frac{96669296395402960755}{298782289485123661444} a + \frac{16274185848600836197}{74695572371280915361}$

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:  \( 29493035.0192 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
$5$5.5.8.8$x^{5} + 15 x^{4} + 5$$5$$1$$8$$F_5$$[2]^{4}$
5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.5.5.3$x^{5} + 15 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$