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Properties

Label 15.15.1070196574...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{21}\cdot 5^{6}\cdot 19^{2}\cdot 37^{5}\cdot 67\cdot 1277^{2}\cdot 5003^{2}\cdot 14779^{4}$
Root discriminant $3430.03$
Ramified primes $2, 5, 19, 37, 67, 1277, 5003, 14779$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-60534784000, -317807616000, -719607244800, -916912806400, -719153233920, -356718963840, -110614717056, -20335308576, -1905031872, -27856440, 9800992, 591772, -11088, -1386, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1386*x^13 - 11088*x^12 + 591772*x^11 + 9800992*x^10 - 27856440*x^9 - 1905031872*x^8 - 20335308576*x^7 - 110614717056*x^6 - 356718963840*x^5 - 719153233920*x^4 - 916912806400*x^3 - 719607244800*x^2 - 317807616000*x - 60534784000)
 
gp: K = bnfinit(x^15 - 1386*x^13 - 11088*x^12 + 591772*x^11 + 9800992*x^10 - 27856440*x^9 - 1905031872*x^8 - 20335308576*x^7 - 110614717056*x^6 - 356718963840*x^5 - 719153233920*x^4 - 916912806400*x^3 - 719607244800*x^2 - 317807616000*x - 60534784000, 1)
 

Normalized defining polynomial

\( x^{15} - 1386 x^{13} - 11088 x^{12} + 591772 x^{11} + 9800992 x^{10} - 27856440 x^{9} - 1905031872 x^{8} - 20335308576 x^{7} - 110614717056 x^{6} - 356718963840 x^{5} - 719153233920 x^{4} - 916912806400 x^{3} - 719607244800 x^{2} - 317807616000 x - 60534784000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(107019657496813016739558827186067295629990920192000000=2^{21}\cdot 5^{6}\cdot 19^{2}\cdot 37^{5}\cdot 67\cdot 1277^{2}\cdot 5003^{2}\cdot 14779^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3430.03$
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:  $2, 5, 19, 37, 67, 1277, 5003, 14779$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{40} a^{5} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} + \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} - \frac{9}{200} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} + \frac{3}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{9}{800} a^{5} + \frac{1}{50} a^{4} + \frac{37}{400} a^{3} - \frac{1}{20} a$, $\frac{1}{12800} a^{10} - \frac{1}{6400} a^{9} - \frac{1}{6400} a^{8} - \frac{11}{3200} a^{7} - \frac{1}{128} a^{6} - \frac{11}{1600} a^{5} + \frac{177}{1600} a^{4} - \frac{29}{800} a^{3} - \frac{89}{400} a^{2} - \frac{9}{40} a + \frac{1}{10}$, $\frac{1}{256000} a^{11} - \frac{3}{128000} a^{9} + \frac{7}{16000} a^{8} + \frac{273}{64000} a^{7} - \frac{9}{8000} a^{6} - \frac{17}{6400} a^{5} + \frac{107}{4000} a^{4} - \frac{939}{4000} a^{3} + \frac{433}{2000} a^{2} + \frac{133}{400} a - \frac{9}{100}$, $\frac{1}{2560000} a^{12} + \frac{17}{1280000} a^{10} - \frac{9}{80000} a^{9} - \frac{387}{640000} a^{8} - \frac{81}{20000} a^{7} - \frac{49}{12800} a^{6} - \frac{11}{5000} a^{5} - \frac{667}{20000} a^{4} + \frac{17}{5000} a^{3} + \frac{51}{4000} a^{2} + \frac{183}{500} a - \frac{11}{50}$, $\frac{1}{5120000000} a^{13} + \frac{19}{128000000} a^{12} - \frac{4813}{2560000000} a^{11} + \frac{543}{80000000} a^{10} + \frac{74783}{1280000000} a^{9} + \frac{37963}{80000000} a^{8} + \frac{7973}{5120000} a^{7} - \frac{5823}{5000000} a^{6} - \frac{641253}{160000000} a^{5} + \frac{2667513}{40000000} a^{4} - \frac{1723529}{8000000} a^{3} + \frac{149731}{1000000} a^{2} - \frac{154121}{400000} a + \frac{28777}{100000}$, $\frac{1}{655360000000} a^{14} - \frac{11}{163840000000} a^{13} + \frac{59667}{327680000000} a^{12} + \frac{66757}{81920000000} a^{11} + \frac{5379631}{163840000000} a^{10} + \frac{715469}{40960000000} a^{9} - \frac{2531391}{81920000000} a^{8} + \frac{47267039}{20480000000} a^{7} - \frac{201692109}{20480000000} a^{6} + \frac{30580933}{2560000000} a^{5} - \frac{368318097}{5120000000} a^{4} + \frac{22513791}{256000000} a^{3} + \frac{63610947}{256000000} a^{2} + \frac{401299}{6400000} a - \frac{174677}{3200000}$

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:  $14$
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:  \( 88308692331100000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.3$x^{6} + 2 x^{2} + 14$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$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$$[\ ]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
37Data not computed
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
1277Data not computed
5003Data not computed
14779Data not computed