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Properties

Label 15.11.2046079521...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{15}\cdot 3^{21}\cdot 5^{6}\cdot 17^{4}\cdot 127^{4}\cdot 1709^{2}\cdot 7759^{2}$
Root discriminant $1222.91$
Ramified primes $2, 3, 5, 17, 127, 1709, 7759$
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![-8843264000, -46427136000, -105124300800, -133947814400, -105057976320, -52098560640, -16113845376, -2907652896, -234539712, 11537640, 4049952, 237732, -8208, -1026, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1026*x^13 - 8208*x^12 + 237732*x^11 + 4049952*x^10 + 11537640*x^9 - 234539712*x^8 - 2907652896*x^7 - 16113845376*x^6 - 52098560640*x^5 - 105057976320*x^4 - 133947814400*x^3 - 105124300800*x^2 - 46427136000*x - 8843264000)
 
gp: K = bnfinit(x^15 - 1026*x^13 - 8208*x^12 + 237732*x^11 + 4049952*x^10 + 11537640*x^9 - 234539712*x^8 - 2907652896*x^7 - 16113845376*x^6 - 52098560640*x^5 - 105057976320*x^4 - 133947814400*x^3 - 105124300800*x^2 - 46427136000*x - 8843264000, 1)
 

Normalized defining polynomial

\( x^{15} - 1026 x^{13} - 8208 x^{12} + 237732 x^{11} + 4049952 x^{10} + 11537640 x^{9} - 234539712 x^{8} - 2907652896 x^{7} - 16113845376 x^{6} - 52098560640 x^{5} - 105057976320 x^{4} - 133947814400 x^{3} - 105124300800 x^{2} - 46427136000 x - 8843264000 \)

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:  \(20460795210884509800211491460024203449856000000=2^{15}\cdot 3^{21}\cdot 5^{6}\cdot 17^{4}\cdot 127^{4}\cdot 1709^{2}\cdot 7759^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1222.91$
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:  $2, 3, 5, 17, 127, 1709, 7759$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{5} - \frac{1}{20} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{160} a^{6} - \frac{1}{80} a^{5} + \frac{1}{20} a^{4} + \frac{1}{20} a^{3} + \frac{3}{40} a^{2} - \frac{1}{4} a$, $\frac{1}{160} a^{7} + \frac{1}{40} a^{4} - \frac{1}{40} a^{3} + \frac{1}{20} a^{2}$, $\frac{1}{1600} a^{8} - \frac{1}{400} a^{6} + \frac{1}{200} a^{5} + \frac{21}{400} a^{4} + \frac{9}{100} a^{3} + \frac{4}{25} a^{2} + \frac{2}{5}$, $\frac{1}{6400} a^{9} + \frac{3}{3200} a^{7} + \frac{1}{800} a^{6} - \frac{19}{1600} a^{5} + \frac{3}{50} a^{4} - \frac{83}{800} a^{3} + \frac{9}{40} a^{2} + \frac{9}{40} a - \frac{1}{2}$, $\frac{1}{25600} a^{10} - \frac{1}{12800} a^{9} + \frac{3}{12800} a^{8} + \frac{9}{6400} a^{7} + \frac{17}{6400} a^{6} - \frac{13}{3200} a^{5} - \frac{99}{3200} a^{4} - \frac{17}{1600} a^{3} + \frac{27}{160} a^{2} - \frac{29}{80} a - \frac{1}{4}$, $\frac{1}{512000} a^{11} + \frac{17}{256000} a^{9} - \frac{3}{32000} a^{8} - \frac{117}{128000} a^{7} + \frac{21}{16000} a^{6} + \frac{11}{2560} a^{5} + \frac{337}{8000} a^{4} + \frac{861}{8000} a^{3} - \frac{137}{4000} a^{2} - \frac{27}{800} a + \frac{51}{200}$, $\frac{1}{10240000} a^{12} - \frac{1}{1024000} a^{11} + \frac{97}{5120000} a^{10} + \frac{63}{2560000} a^{9} + \frac{723}{2560000} a^{8} - \frac{3091}{1280000} a^{7} + \frac{799}{256000} a^{6} - \frac{7067}{640000} a^{5} - \frac{6609}{160000} a^{4} + \frac{1049}{40000} a^{3} + \frac{659}{3200} a^{2} - \frac{843}{8000} a + \frac{37}{400}$, $\frac{1}{10240000000} a^{13} - \frac{3}{128000000} a^{12} + \frac{367}{5120000000} a^{11} + \frac{7327}{640000000} a^{10} + \frac{128873}{2560000000} a^{9} + \frac{21781}{320000000} a^{8} - \frac{730343}{256000000} a^{7} - \frac{145003}{160000000} a^{6} + \frac{1203687}{320000000} a^{5} - \frac{4183127}{80000000} a^{4} + \frac{684491}{16000000} a^{3} + \frac{83169}{500000} a^{2} + \frac{139559}{800000} a + \frac{9217}{200000}$, $\frac{1}{2621440000000} a^{14} - \frac{11}{655360000000} a^{13} + \frac{59847}{1310720000000} a^{12} - \frac{254863}{327680000000} a^{11} - \frac{9475759}{655360000000} a^{10} + \frac{2912839}{163840000000} a^{9} + \frac{27873589}{327680000000} a^{8} + \frac{201359459}{81920000000} a^{7} + \frac{247415011}{81920000000} a^{6} - \frac{96296357}{10240000000} a^{5} - \frac{1055707437}{20480000000} a^{4} - \frac{71492189}{1024000000} a^{3} - \frac{91676713}{1024000000} a^{2} + \frac{2975079}{25600000} a - \frac{1643617}{12800000}$

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:  \( 1866318824730000000 \) (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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $15$ $15$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$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}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.5.4.1$x^{5} - 17$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
127Data not computed
1709Data not computed
7759Data not computed