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Properties

Label 15.11.1321365067...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{15}\cdot 3^{24}\cdot 5^{6}\cdot 53^{4}\cdot 1783^{2}\cdot 603557^{2}$
Root discriminant $1018.75$
Ramified primes $2, 3, 5, 53, 1783, 603557$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![651264000, 0, -2466662400, -632947200, 2439797760, 1226545920, -55931904, -120731040, -18133632, 281880, 273216, 34944, -864, -378, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 378*x^13 - 864*x^12 + 34944*x^11 + 273216*x^10 + 281880*x^9 - 18133632*x^8 - 120731040*x^7 - 55931904*x^6 + 1226545920*x^5 + 2439797760*x^4 - 632947200*x^3 - 2466662400*x^2 + 651264000)
 
gp: K = bnfinit(x^15 - 378*x^13 - 864*x^12 + 34944*x^11 + 273216*x^10 + 281880*x^9 - 18133632*x^8 - 120731040*x^7 - 55931904*x^6 + 1226545920*x^5 + 2439797760*x^4 - 632947200*x^3 - 2466662400*x^2 + 651264000, 1)
 

Normalized defining polynomial

\( x^{15} - 378 x^{13} - 864 x^{12} + 34944 x^{11} + 273216 x^{10} + 281880 x^{9} - 18133632 x^{8} - 120731040 x^{7} - 55931904 x^{6} + 1226545920 x^{5} + 2439797760 x^{4} - 632947200 x^{3} - 2466662400 x^{2} + 651264000 \)

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:  \(1321365067721650998272614708394954813952000000=2^{15}\cdot 3^{24}\cdot 5^{6}\cdot 53^{4}\cdot 1783^{2}\cdot 603557^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1018.75$
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:  $2, 3, 5, 53, 1783, 603557$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{32} a^{7} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{256} a^{10} - \frac{1}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{64} a^{7} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} - \frac{1}{32} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{2560} a^{11} - \frac{9}{1280} a^{9} - \frac{3}{320} a^{8} + \frac{1}{40} a^{7} + \frac{1}{160} a^{6} + \frac{3}{64} a^{5} - \frac{3}{40} a^{4} - \frac{3}{16} a^{3} - \frac{1}{40} a^{2} - \frac{1}{4} a$, $\frac{1}{5120} a^{12} + \frac{1}{2560} a^{10} + \frac{1}{320} a^{9} - \frac{7}{640} a^{8} - \frac{1}{80} a^{7} + \frac{7}{128} a^{6} - \frac{3}{80} a^{5} - \frac{1}{8} a^{4} + \frac{7}{40} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{51200} a^{13} + \frac{1}{25600} a^{11} - \frac{1}{800} a^{10} + \frac{13}{6400} a^{9} + \frac{3}{1600} a^{8} + \frac{3}{256} a^{7} - \frac{43}{800} a^{6} - \frac{1}{80} a^{5} + \frac{3}{100} a^{4} - \frac{17}{80} a^{3} - \frac{7}{40} a^{2}$, $\frac{1}{2397571327315108237508878334906246767880396800} a^{14} + \frac{116464373567182258855773228818912365767}{18731025994649283105538111991455052874065600} a^{13} - \frac{15904807730378739995216783539915288976349}{1198785663657554118754439167453123383940198400} a^{12} + \frac{4910946910427759377500471575120779820723}{149848207957194264844304895931640422992524800} a^{11} - \frac{2152963224194840023077647396074756860047}{1873102599464928310553811199145505287406560} a^{10} + \frac{88043191384890364734165428529140787341761}{14984820795719426484430489593164042299252480} a^{9} + \frac{3696652248358048874488324530279890633744147}{299696415914388529688609791863280845985049600} a^{8} + \frac{455560600432720693276313372305864643844281}{18731025994649283105538111991455052874065600} a^{7} + \frac{4267085231221164566810482686878996412696059}{74924103978597132422152447965820211496262400} a^{6} + \frac{1534744593505112126167325839751572750428007}{18731025994649283105538111991455052874065600} a^{5} + \frac{1067084198297385632195388123657222019254191}{9365512997324641552769055995727526437032800} a^{4} - \frac{80909851572438292084856650702462132533659}{936551299732464155276905599572752643703280} a^{3} - \frac{230420299535253729004935701171522270256323}{936551299732464155276905599572752643703280} a^{2} - \frac{3997610073177286389031565825607473772713}{23413782493311603881922639989318816092582} a - \frac{3682008220000687017328229459524511619757}{11706891246655801940961319994659408046291}$

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:  \( 476218915504000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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 sibling: 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ R ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53Data not computed
1783Data not computed
603557Data not computed