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Properties

Label 15.7.47608134035...5361.1
Degree $15$
Signature $[7, 4]$
Discriminant $61^{4}\cdot 139^{2}\cdot 397^{4}\cdot 267661^{2}$
Root discriminant $150.84$
Ramified primes $61, 139, 397, 267661$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T89

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2598643, -6186220, -7914792, -5503034, -1857141, 359939, 579191, 188292, -28869, -28992, -3072, 2056, 271, -75, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 75*x^13 + 271*x^12 + 2056*x^11 - 3072*x^10 - 28992*x^9 - 28869*x^8 + 188292*x^7 + 579191*x^6 + 359939*x^5 - 1857141*x^4 - 5503034*x^3 - 7914792*x^2 - 6186220*x - 2598643)
 
gp: K = bnfinit(x^15 - 5*x^14 - 75*x^13 + 271*x^12 + 2056*x^11 - 3072*x^10 - 28992*x^9 - 28869*x^8 + 188292*x^7 + 579191*x^6 + 359939*x^5 - 1857141*x^4 - 5503034*x^3 - 7914792*x^2 - 6186220*x - 2598643, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 75 x^{13} + 271 x^{12} + 2056 x^{11} - 3072 x^{10} - 28992 x^{9} - 28869 x^{8} + 188292 x^{7} + 579191 x^{6} + 359939 x^{5} - 1857141 x^{4} - 5503034 x^{3} - 7914792 x^{2} - 6186220 x - 2598643 \)

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:  \(476081340356745953854630586535361=61^{4}\cdot 139^{2}\cdot 397^{4}\cdot 267661^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.84$
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:  $61, 139, 397, 267661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{6}{13} a^{11} + \frac{1}{13} a^{10} - \frac{6}{13} a^{9} + \frac{3}{13} a^{8} - \frac{2}{13} a^{7} + \frac{1}{13} a^{6} + \frac{4}{13} a^{5} - \frac{4}{13} a^{4} + \frac{1}{13} a^{3} + \frac{4}{13} a^{2} + \frac{3}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{13} + \frac{4}{13} a^{11} + \frac{1}{13} a^{10} + \frac{6}{13} a^{8} - \frac{2}{13} a^{6} - \frac{2}{13} a^{5} - \frac{1}{13} a^{4} - \frac{2}{13} a^{3} + \frac{5}{13} a^{2} + \frac{2}{13} a - \frac{3}{13}$, $\frac{1}{171288685689004148332368663312197} a^{14} - \frac{6405572353332639823579013522870}{171288685689004148332368663312197} a^{13} + \frac{2381568148423421128096608822851}{171288685689004148332368663312197} a^{12} + \frac{79002009146351238202908970597885}{171288685689004148332368663312197} a^{11} - \frac{81226514511386409584993757575933}{171288685689004148332368663312197} a^{10} - \frac{42681566077371714038076842650044}{171288685689004148332368663312197} a^{9} - \frac{58831936103456657228941575712082}{171288685689004148332368663312197} a^{8} + \frac{20954000343290767081772234985283}{171288685689004148332368663312197} a^{7} - \frac{70463695119345487297980655157658}{171288685689004148332368663312197} a^{6} + \frac{4127469047942876215546045373787}{171288685689004148332368663312197} a^{5} - \frac{76525426936662489005479943271977}{171288685689004148332368663312197} a^{4} - \frac{215932000427497102303201309836}{13176052745308011410182204870169} a^{3} + \frac{50809541216660288691160591421760}{171288685689004148332368663312197} a^{2} - \frac{6341912203879477354378385798380}{13176052745308011410182204870169} a + \frac{15654867151679184384238317914171}{171288685689004148332368663312197}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 19135723147.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T89:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 60 conjugacy class representatives for 1/2[S(3)^5]S(5) are not computed
Character table for 1/2[S(3)^5]S(5) is not computed

Intermediate fields

5.5.24217.1

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

Sibling fields

Degree 30 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 $15$ $15$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
61Data not computed
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.3.2.2$x^{3} + 556$$3$$1$$2$$C_3$$[\ ]_{3}$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.6.0.1$x^{6} - x + 21$$1$$6$$0$$C_6$$[\ ]^{6}$
397Data not computed
267661Data not computed