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Properties

Label 15.3.69512094038...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{18}\cdot 3^{4}\cdot 5^{6}\cdot 13^{2}\cdot 31^{5}\cdot 208094267^{2}$
Root discriminant $333.28$
Ramified primes $2, 3, 5, 13, 31, 208094267$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 15T97

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-331776000, 580608000, -298598400, 359769600, -311662080, 48684160, -77250944, 15375488, 1975680, 488376, -37888, 15904, -432, 234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 234*x^13 - 432*x^12 + 15904*x^11 - 37888*x^10 + 488376*x^9 + 1975680*x^8 + 15375488*x^7 - 77250944*x^6 + 48684160*x^5 - 311662080*x^4 + 359769600*x^3 - 298598400*x^2 + 580608000*x - 331776000)
 
gp: K = bnfinit(x^15 + 234*x^13 - 432*x^12 + 15904*x^11 - 37888*x^10 + 488376*x^9 + 1975680*x^8 + 15375488*x^7 - 77250944*x^6 + 48684160*x^5 - 311662080*x^4 + 359769600*x^3 - 298598400*x^2 + 580608000*x - 331776000, 1)
 

Normalized defining polynomial

\( x^{15} + 234 x^{13} - 432 x^{12} + 15904 x^{11} - 37888 x^{10} + 488376 x^{9} + 1975680 x^{8} + 15375488 x^{7} - 77250944 x^{6} + 48684160 x^{5} - 311662080 x^{4} + 359769600 x^{3} - 298598400 x^{2} + 580608000 x - 331776000 \)

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:  \(69512094038728469161467479126016000000=2^{18}\cdot 3^{4}\cdot 5^{6}\cdot 13^{2}\cdot 31^{5}\cdot 208094267^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $333.28$
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:  $2, 3, 5, 13, 31, 208094267$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{32} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{10} - \frac{1}{64} a^{8} - \frac{1}{16} a^{6} - \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1280} a^{11} - \frac{3}{640} a^{9} + \frac{1}{160} a^{8} - \frac{1}{80} a^{7} - \frac{3}{80} a^{6} - \frac{13}{160} a^{5} + \frac{1}{10} a^{3} - \frac{1}{20} a^{2}$, $\frac{1}{7680} a^{12} - \frac{1}{1280} a^{10} + \frac{1}{160} a^{9} + \frac{1}{120} a^{8} + \frac{1}{240} a^{7} - \frac{11}{320} a^{6} - \frac{1}{8} a^{5} + \frac{1}{60} a^{4} - \frac{13}{60} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{460800} a^{13} - \frac{7}{25600} a^{11} - \frac{3}{3200} a^{10} - \frac{221}{28800} a^{9} + \frac{19}{3600} a^{8} - \frac{137}{6400} a^{7} + \frac{1}{80} a^{6} - \frac{373}{7200} a^{5} + \frac{377}{3600} a^{4} - \frac{53}{720} a^{3} - \frac{1}{20} a^{2} - \frac{1}{4} a$, $\frac{1}{4942070010533595298765870761289271901431193600} a^{14} + \frac{1024091766407334169191212609131389209}{17159965314352761454048162365587749657747200} a^{13} + \frac{13262888955792363341842292385799091848013}{274559445029644183264770597849403994523955200} a^{12} - \frac{2599472724770955974135611590447724905627}{11439976876235174302698774910391833105164800} a^{11} + \frac{111644742993081478106083908201761010850741}{30887937565834970617286692258057949383944960} a^{10} + \frac{17948450708213176904433007392640280641387}{15443968782917485308643346129028974691972480} a^{9} - \frac{22423355498920312493488505839435722488137}{2745594450296441832647705978494039945239552} a^{8} + \frac{49609959913614370757476517090292609377641}{4289991328588190363512040591396937414436800} a^{7} - \frac{937313740642701130967767253464507597914899}{38609921957293713271608365322572436729931200} a^{6} + \frac{1688712862692523473341092540790347268706839}{38609921957293713271608365322572436729931200} a^{5} - \frac{461542018530617255382565690645555521207271}{38609921957293713271608365322572436729931200} a^{4} - \frac{16844209143574239118382245668634482248257}{71499855476469839391867343189948956907280} a^{3} - \frac{21683926161932363463463261805233948694533}{214499566429409518175602029569846870721840} a^{2} + \frac{13833144684013627586011401268464845019}{893748193455872992398341789874361961341} a + \frac{259619628670395338259143291907087019461}{893748193455872992398341789874361961341}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 10556637500500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(3) is not computed

Intermediate fields

3.1.31.1

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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.10.0.1$x^{10} + 2 x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
208094267Data not computed