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Properties

Label 15.3.51525487947...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{16}\cdot 5^{6}\cdot 131^{4}\cdot 733^{5}\cdot 2053^{2}\cdot 437693^{2}$
Root discriminant $2061.27$
Ramified primes $2, 5, 131, 733, 2053, 437693$
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![-4292608000, 26292224000, -55455129600, 46507724800, -12492495360, -3483015040, 2284777472, -23291424, -133337088, -1450872, 1792640, 4208, -3456, -576, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 576*x^13 - 3456*x^12 + 4208*x^11 + 1792640*x^10 - 1450872*x^9 - 133337088*x^8 - 23291424*x^7 + 2284777472*x^6 - 3483015040*x^5 - 12492495360*x^4 + 46507724800*x^3 - 55455129600*x^2 + 26292224000*x - 4292608000)
 
gp: K = bnfinit(x^15 - 576*x^13 - 3456*x^12 + 4208*x^11 + 1792640*x^10 - 1450872*x^9 - 133337088*x^8 - 23291424*x^7 + 2284777472*x^6 - 3483015040*x^5 - 12492495360*x^4 + 46507724800*x^3 - 55455129600*x^2 + 26292224000*x - 4292608000, 1)
 

Normalized defining polynomial

\( x^{15} - 576 x^{13} - 3456 x^{12} + 4208 x^{11} + 1792640 x^{10} - 1450872 x^{9} - 133337088 x^{8} - 23291424 x^{7} + 2284777472 x^{6} - 3483015040 x^{5} - 12492495360 x^{4} + 46507724800 x^{3} - 55455129600 x^{2} + 26292224000 x - 4292608000 \)

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:  \(51525487947813225854981517260718473944855552000000=2^{16}\cdot 5^{6}\cdot 131^{4}\cdot 733^{5}\cdot 2053^{2}\cdot 437693^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2061.27$
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:  $2, 5, 131, 733, 2053, 437693$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{256} a^{10} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{2560} a^{11} + \frac{1}{640} a^{9} + \frac{3}{320} a^{8} + \frac{3}{160} a^{7} - \frac{1}{16} a^{6} + \frac{1}{320} a^{5} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3} + \frac{3}{40} a^{2}$, $\frac{1}{10240} a^{12} - \frac{1}{640} a^{10} - \frac{1}{640} a^{9} - \frac{7}{640} a^{8} + \frac{1}{1280} a^{6} + \frac{1}{20} a^{5} - \frac{37}{320} a^{4} - \frac{1}{80} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{204800} a^{13} - \frac{1}{12800} a^{11} - \frac{1}{800} a^{10} + \frac{3}{12800} a^{9} + \frac{1}{160} a^{8} - \frac{399}{25600} a^{7} + \frac{1}{400} a^{6} + \frac{283}{6400} a^{5} - \frac{17}{200} a^{4} + \frac{13}{320} a^{3} + \frac{1}{80} a^{2}$, $\frac{1}{23210556964050105350624492874076205563118050429337600} a^{14} + \frac{83328041567017269799729266842434724812722669}{2901319620506263168828061609259525695389756303667200} a^{13} - \frac{44739844973665082797655663720429336375878622741}{1450659810253131584414030804629762847694878151833600} a^{12} + \frac{23944798902467067012617786058079013936650472063}{362664952563282896103507701157440711923719537958400} a^{11} - \frac{336492912397110962263399156328576157658248642889}{1450659810253131584414030804629762847694878151833600} a^{10} + \frac{109200830913202954268872137730570489055185845441}{90666238140820724025876925289360177980929884489600} a^{9} + \frac{26072752244604088934797163350646171278453661129841}{2901319620506263168828061609259525695389756303667200} a^{8} + \frac{2022367150197844670675070564576326506758936219117}{362664952563282896103507701157440711923719537958400} a^{7} + \frac{8236426543518868212209948417476898074969039912271}{145065981025313158441403080462976284769487815183360} a^{6} - \frac{3981473081104170419469438472155394723787313332211}{90666238140820724025876925289360177980929884489600} a^{5} - \frac{14258136905507917796645643088945641395331775256507}{181332476281641448051753850578720355961859768979200} a^{4} + \frac{1396669529137189708916314774030091768640538828889}{9066623814082072402587692528936017798092988448960} a^{3} - \frac{327227202297628707338912668822616176572970643}{14437299067009669430872121861363085665753166320} a^{2} + \frac{14687761039958679178256672114016188396471105537}{56666398838012952516173078305850111238081177806} a - \frac{13343923271232060134863831108624488607548902610}{28333199419006476258086539152925055619040588903}$

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:  $8$
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:  \( 192322044966000000000 \) (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

3.3.733.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $15$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.5$x^{4} + 2 x^{2} - 4$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$131$131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.5.4.2$x^{5} + 393$$5$$1$$4$$C_5$$[\ ]_{5}$
131.6.0.1$x^{6} - 3 x + 54$$1$$6$$0$$C_6$$[\ ]^{6}$
733Data not computed
2053Data not computed
437693Data not computed