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Properties

Label 15.7.98539808552...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 5^{6}\cdot 13^{4}\cdot 37^{5}\cdot 3485257483^{2}$
Root discriminant $540.64$
Ramified primes $2, 5, 13, 37, 3485257483$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104000, 364000, -676000, -1561300, 2896920, -460300, -1379644, 978344, -301752, 63459, -15324, 2795, -432, 99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 99*x^13 - 432*x^12 + 2795*x^11 - 15324*x^10 + 63459*x^9 - 301752*x^8 + 978344*x^7 - 1379644*x^6 - 460300*x^5 + 2896920*x^4 - 1561300*x^3 - 676000*x^2 + 364000*x + 104000)
 
gp: K = bnfinit(x^15 + 99*x^13 - 432*x^12 + 2795*x^11 - 15324*x^10 + 63459*x^9 - 301752*x^8 + 978344*x^7 - 1379644*x^6 - 460300*x^5 + 2896920*x^4 - 1561300*x^3 - 676000*x^2 + 364000*x + 104000, 1)
 

Normalized defining polynomial

\( x^{15} + 99 x^{13} - 432 x^{12} + 2795 x^{11} - 15324 x^{10} + 63459 x^{9} - 301752 x^{8} + 978344 x^{7} - 1379644 x^{6} - 460300 x^{5} + 2896920 x^{4} - 1561300 x^{3} - 676000 x^{2} + 364000 x + 104000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98539808552158008427621845251903488000000=2^{18}\cdot 5^{6}\cdot 13^{4}\cdot 37^{5}\cdot 3485257483^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $540.64$
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:  $2, 5, 13, 37, 3485257483$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{1}{10} a^{8} - \frac{1}{4} a^{7} + \frac{3}{10} a^{6} - \frac{1}{20} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{3}{10} a^{2}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{20} a^{9} + \frac{3}{8} a^{8} + \frac{3}{20} a^{7} + \frac{19}{40} a^{6} + \frac{9}{20} a^{5} - \frac{2}{5} a^{4} + \frac{3}{20} a^{3}$, $\frac{1}{200} a^{13} - \frac{1}{200} a^{11} + \frac{9}{100} a^{10} - \frac{1}{40} a^{9} + \frac{13}{100} a^{8} - \frac{41}{200} a^{7} - \frac{1}{100} a^{6} + \frac{11}{50} a^{5} - \frac{47}{100} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{7529977263810116569717283246135194000} a^{14} + \frac{931977213392158967238244963936637}{752997726381011656971728324613519400} a^{13} - \frac{66583979008149303412692017532342731}{7529977263810116569717283246135194000} a^{12} - \frac{5600484541010853583261451048019501}{3764988631905058284858641623067597000} a^{11} - \frac{87002891123440002795028220939030363}{1505995452762023313943456649227038800} a^{10} - \frac{54411686682660902913181576388667957}{3764988631905058284858641623067597000} a^{9} - \frac{463854567603302826581308043868067271}{7529977263810116569717283246135194000} a^{8} - \frac{113780962371852916895265664131080651}{3764988631905058284858641623067597000} a^{7} - \frac{635884666886157744542800970608163033}{3764988631905058284858641623067597000} a^{6} - \frac{385634306623966477781517941787810763}{941247157976264571214660405766899250} a^{5} - \frac{16663412979819844592025548090410061}{75299772638101165697172832461351940} a^{4} + \frac{98220805758349192538382856211096067}{376498863190505828485864162306759700} a^{3} + \frac{7236768252774274913513408398123335}{15059954527620233139434566492270388} a^{2} - \frac{1019785735457923913946817450588711}{37649886319050582848586416230675970} a + \frac{1095601117771456134691416737933625}{3764988631905058284858641623067597}$

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:  $10$
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:  \( 8955971917100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]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 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$[\ ]_{3}^{2}$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.10.1$x^{6} + 2 x^{5} + 2 x^{4} + 2$$6$$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$$[\ ]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
37Data not computed
3485257483Data not computed