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Properties

Label 15.11.6242427558...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 19^{10}\cdot 37^{4}\cdot 1481^{2}\cdot 158699^{2}$
Root discriminant $1790.70$
Ramified primes $2, 5, 7, 19, 37, 1481, 158699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1060864000, -5569536000, -12611020800, -16068774400, -12603064320, -6255936640, -1954203776, -378179296, -48520512, -5886360, -737248, -47968, -1008, -126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000)
 
gp: K = bnfinit(x^15 - 126*x^13 - 1008*x^12 - 47968*x^11 - 737248*x^10 - 5886360*x^9 - 48520512*x^8 - 378179296*x^7 - 1954203776*x^6 - 6255936640*x^5 - 12603064320*x^4 - 16068774400*x^3 - 12611020800*x^2 - 5569536000*x - 1060864000, 1)
 

Normalized defining polynomial

\( x^{15} - 126 x^{13} - 1008 x^{12} - 47968 x^{11} - 737248 x^{10} - 5886360 x^{9} - 48520512 x^{8} - 378179296 x^{7} - 1954203776 x^{6} - 6255936640 x^{5} - 12603064320 x^{4} - 16068774400 x^{3} - 12611020800 x^{2} - 5569536000 x - 1060864000 \)

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:  \(6242427558646320395593447809832981520060416000000=2^{18}\cdot 5^{6}\cdot 7^{4}\cdot 19^{10}\cdot 37^{4}\cdot 1481^{2}\cdot 158699^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1790.70$
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:  $2, 5, 7, 19, 37, 1481, 158699$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} - \frac{7}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{1}{200} a^{5} + \frac{3}{25} a^{4} + \frac{17}{400} a^{3} + \frac{1}{10} a^{2} + \frac{9}{20} a$, $\frac{1}{6400} a^{10} + \frac{1}{3200} a^{8} - \frac{1}{200} a^{7} - \frac{1}{80} a^{6} - \frac{1}{50} a^{5} - \frac{3}{160} a^{4} + \frac{11}{100} a^{3} - \frac{7}{200} a^{2} - \frac{2}{5}$, $\frac{1}{128000} a^{11} + \frac{7}{64000} a^{9} - \frac{3}{8000} a^{8} + \frac{89}{16000} a^{7} - \frac{9}{4000} a^{6} - \frac{27}{3200} a^{5} - \frac{83}{2000} a^{4} + \frac{153}{1000} a^{3} - \frac{87}{1000} a^{2} - \frac{97}{200} a + \frac{1}{50}$, $\frac{1}{1280000} a^{12} + \frac{47}{640000} a^{10} + \frac{3}{20000} a^{9} - \frac{11}{160000} a^{8} - \frac{97}{20000} a^{7} + \frac{277}{32000} a^{6} - \frac{323}{20000} a^{5} - \frac{341}{5000} a^{4} - \frac{117}{625} a^{3} - \frac{249}{2000} a^{2} + \frac{34}{125} a + \frac{9}{25}$, $\frac{1}{5120000000} a^{13} - \frac{31}{128000000} a^{12} + \frac{5817}{2560000000} a^{11} + \frac{451}{160000000} a^{10} - \frac{2993}{20000000} a^{9} - \frac{36319}{160000000} a^{8} - \frac{148343}{128000000} a^{7} - \frac{75207}{20000000} a^{6} + \frac{2239237}{160000000} a^{5} + \frac{717073}{40000000} a^{4} - \frac{1208209}{8000000} a^{3} - \frac{130799}{1000000} a^{2} + \frac{181959}{400000} a + \frac{31017}{100000}$, $\frac{1}{655360000000} a^{14} - \frac{11}{163840000000} a^{13} + \frac{60297}{327680000000} a^{12} - \frac{258913}{81920000000} a^{11} - \frac{63003}{1280000000} a^{10} + \frac{2478657}{20480000000} a^{9} - \frac{32003811}{81920000000} a^{8} + \frac{30777559}{20480000000} a^{7} - \frac{198911339}{20480000000} a^{6} + \frac{10367617}{640000000} a^{5} + \frac{39834263}{5120000000} a^{4} - \frac{1347289}{256000000} a^{3} - \frac{33023413}{256000000} a^{2} - \frac{1598821}{6400000} a - \frac{1323917}{3200000}$

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:  \( 33525688191400000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

3.3.361.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]^{3}$
2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$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}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
19Data not computed
37Data not computed
1481Data not computed
158699Data not computed