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Properties

Label 15.9.23640046139...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{20}\cdot 5^{6}\cdot 37^{5}\cdot 113^{4}\cdot 139^{2}\cdot 257003^{2}$
Root discriminant $573.12$
Ramified primes $2, 5, 37, 113, 139, 257003$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![904000, 3164000, -5876000, -13571300, 25180920, -4794300, -8858444, 4223644, 225648, -483381, 95676, -2395, -792, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 9*x^13 - 792*x^12 - 2395*x^11 + 95676*x^10 - 483381*x^9 + 225648*x^8 + 4223644*x^7 - 8858444*x^6 - 4794300*x^5 + 25180920*x^4 - 13571300*x^3 - 5876000*x^2 + 3164000*x + 904000)
 
gp: K = bnfinit(x^15 + 9*x^13 - 792*x^12 - 2395*x^11 + 95676*x^10 - 483381*x^9 + 225648*x^8 + 4223644*x^7 - 8858444*x^6 - 4794300*x^5 + 25180920*x^4 - 13571300*x^3 - 5876000*x^2 + 3164000*x + 904000, 1)
 

Normalized defining polynomial

\( x^{15} + 9 x^{13} - 792 x^{12} - 2395 x^{11} + 95676 x^{10} - 483381 x^{9} + 225648 x^{8} + 4223644 x^{7} - 8858444 x^{6} - 4794300 x^{5} + 25180920 x^{4} - 13571300 x^{3} - 5876000 x^{2} + 3164000 x + 904000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-236400461395948294457915703984996352000000=-\,2^{20}\cdot 5^{6}\cdot 37^{5}\cdot 113^{4}\cdot 139^{2}\cdot 257003^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $573.12$
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:  $2, 5, 37, 113, 139, 257003$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{1}{10} a^{8} - \frac{1}{4} a^{7} + \frac{3}{10} a^{6} + \frac{9}{20} a^{5} - \frac{1}{10} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} + \frac{3}{10} a^{7} + \frac{9}{20} a^{6} - \frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3}$, $\frac{1}{200} a^{13} - \frac{1}{200} a^{11} + \frac{1}{25} a^{10} + \frac{3}{40} a^{9} + \frac{12}{25} a^{8} + \frac{69}{200} a^{7} - \frac{3}{50} a^{6} - \frac{23}{100} a^{5} - \frac{3}{25} a^{4} - \frac{1}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{1382248138785721082672091970860053545366000} a^{14} + \frac{212310774823324211329690736451885249837}{138224813878572108267209197086005354536600} a^{13} - \frac{33974145839025689462655481752861460875171}{1382248138785721082672091970860053545366000} a^{12} + \frac{13807975602883254587980355534097907130419}{691124069392860541336045985430026772683000} a^{11} + \frac{2745544531578971792246105495566456196669}{276449627757144216534418394172010709073200} a^{10} - \frac{145356118548290913341610636830892864453107}{691124069392860541336045985430026772683000} a^{9} + \frac{88608897328354221865103135618722520254839}{1382248138785721082672091970860053545366000} a^{8} - \frac{159363859666825260911593958424303371920651}{691124069392860541336045985430026772683000} a^{7} - \frac{68926901733472692337417431391181992724527}{172781017348215135334011496357506693170750} a^{6} - \frac{24813726550871274795087808426835463915713}{172781017348215135334011496357506693170750} a^{5} + \frac{536825755379900149796308511958885709731}{2764496277571442165344183941720107090732} a^{4} + \frac{8016644710529005154170023035565829281781}{34556203469643027066802299271501338634150} a^{3} - \frac{73138825353570012366815342037668912407}{2764496277571442165344183941720107090732} a^{2} + \frac{2966492598105594095150391302597521612099}{6911240693928605413360459854300267726830} a - \frac{227336052942189455017583672243186244627}{691124069392860541336045985430026772683}$

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:  $11$
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:  \( 32377981625800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(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 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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.8.4$x^{6} + 2 x^{3} + 2 x^{2} + 2$$6$$1$$8$$S_4\times C_2$$[4/3, 4/3, 2]_{3}^{2}$
2.6.10.4$x^{6} + 2 x^{5} + 2 x^{4} + 6$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37Data not computed
113Data not computed
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
257003Data not computed