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Properties

Label 15.11.1593894329...1088.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{14}\cdot 37^{5}\cdot 252949^{2}\cdot 4682551^{2}$
Root discriminant $259.12$
Ramified primes $2, 37, 252949, 4682551$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32768, 204800, -2539520, 3744000, 64640, 980608, 345120, -276240, -36416, -6038, -1774, 2460, 222, -98, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 98*x^13 + 222*x^12 + 2460*x^11 - 1774*x^10 - 6038*x^9 - 36416*x^8 - 276240*x^7 + 345120*x^6 + 980608*x^5 + 64640*x^4 + 3744000*x^3 - 2539520*x^2 + 204800*x + 32768)
 
gp: K = bnfinit(x^15 - 4*x^14 - 98*x^13 + 222*x^12 + 2460*x^11 - 1774*x^10 - 6038*x^9 - 36416*x^8 - 276240*x^7 + 345120*x^6 + 980608*x^5 + 64640*x^4 + 3744000*x^3 - 2539520*x^2 + 204800*x + 32768, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 98 x^{13} + 222 x^{12} + 2460 x^{11} - 1774 x^{10} - 6038 x^{9} - 36416 x^{8} - 276240 x^{7} + 345120 x^{6} + 980608 x^{5} + 64640 x^{4} + 3744000 x^{3} - 2539520 x^{2} + 204800 x + 32768 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1593894329827244387250162786359001088=2^{14}\cdot 37^{5}\cdot 252949^{2}\cdot 4682551^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $259.12$
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:  $2, 37, 252949, 4682551$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{48} a^{9} + \frac{1}{24} a^{8} - \frac{1}{8} a^{7} + \frac{5}{24} a^{6} - \frac{1}{6} a^{5} + \frac{3}{8} a^{4} - \frac{5}{24} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{96} a^{10} + \frac{1}{48} a^{8} - \frac{1}{48} a^{7} - \frac{1}{24} a^{6} + \frac{5}{48} a^{5} - \frac{23}{48} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{192} a^{11} - \frac{1}{96} a^{9} - \frac{5}{96} a^{8} + \frac{5}{48} a^{7} - \frac{5}{32} a^{6} - \frac{7}{96} a^{5} + \frac{1}{24} a^{4} - \frac{1}{4} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{768} a^{12} - \frac{1}{384} a^{10} + \frac{1}{128} a^{9} - \frac{11}{192} a^{8} + \frac{11}{128} a^{7} - \frac{23}{384} a^{6} - \frac{13}{32} a^{5} - \frac{7}{16} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{3072} a^{13} - \frac{1}{1536} a^{11} - \frac{5}{1536} a^{10} - \frac{1}{256} a^{9} + \frac{49}{1536} a^{8} + \frac{89}{1536} a^{7} + \frac{3}{128} a^{6} + \frac{49}{192} a^{5} + \frac{17}{96} a^{4} + \frac{7}{24} a^{3} - \frac{1}{24} a^{2} + \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{1041516553745985857800602751709184} a^{14} + \frac{5298072073300960730883101853}{86793046145498821483383562642432} a^{13} + \frac{324501192095810716008548366807}{520758276872992928900301375854592} a^{12} + \frac{397699487985617399051214542351}{520758276872992928900301375854592} a^{11} - \frac{346077946512404661126660813905}{260379138436496464450150687927296} a^{10} - \frac{448601355458910233606110692525}{173586092290997642966767125284864} a^{9} + \frac{8517214265355852880981619257141}{520758276872992928900301375854592} a^{8} + \frac{2462173276904400540476625482581}{32547392304562058056268835990912} a^{7} - \frac{7965170774265196139943228386767}{65094784609124116112537671981824} a^{6} - \frac{2108320354973069566612811800265}{10849130768187352685422945330304} a^{5} + \frac{2265530840116850066415719205331}{8136848076140514514067208997728} a^{4} - \frac{2386930768741221732764753059579}{8136848076140514514067208997728} a^{3} + \frac{566203780797954709966391012343}{1356141346023419085677868166288} a^{2} - \frac{143536407371368565695788834325}{508553004758782157129200562358} a - \frac{95758717684713657447522330532}{254276502379391078564600281179}$

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:  $12$
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:  \( 80704179128700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T60:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 40 conjugacy class representatives for [D(5)^3]S(3)=D(5)wrS(3)
Character table for [D(5)^3]S(3)=D(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 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
37Data not computed
252949Data not computed
4682551Data not computed