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Properties

Label 15.9.63812621058...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{24}\cdot 5^{6}\cdot 13^{4}\cdot 17\cdot 19^{4}\cdot 37^{5}\cdot 74483599^{2}$
Root discriminant $1131.52$
Ramified primes $2, 5, 13, 17, 19, 37, 74483599$
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![247000, 2593500, 11744850, 29930225, 46949760, 46578665, 28900022, 10660124, 2044656, 97245, -27974, -3868, -396, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000)
 
gp: K = bnfinit(x^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000, 1)
 

Normalized defining polynomial

\( x^{15} - 99 x^{13} - 396 x^{12} - 3868 x^{11} - 27974 x^{10} + 97245 x^{9} + 2044656 x^{8} + 10660124 x^{7} + 28900022 x^{6} + 46578665 x^{5} + 46949760 x^{4} + 29930225 x^{3} + 11744850 x^{2} + 2593500 x + 247000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6381262105849605201341604346805613756416000000=-\,2^{24}\cdot 5^{6}\cdot 13^{4}\cdot 17\cdot 19^{4}\cdot 37^{5}\cdot 74483599^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1131.52$
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:  $2, 5, 13, 17, 19, 37, 74483599$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} - \frac{1}{25} a^{4} - \frac{8}{25} a^{3} + \frac{4}{25} a^{2} - \frac{2}{5}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{1}{25} a^{6} - \frac{1}{25} a^{5} - \frac{8}{25} a^{4} + \frac{4}{25} a^{3} - \frac{2}{5} a$, $\frac{1}{25} a^{10} + \frac{1}{25} a^{7} - \frac{2}{25} a^{6} - \frac{2}{25} a^{5} + \frac{3}{25} a^{4} + \frac{2}{25} a^{3} + \frac{9}{25} a^{2} - \frac{2}{5}$, $\frac{1}{125} a^{11} + \frac{1}{125} a^{9} - \frac{1}{125} a^{8} - \frac{3}{125} a^{7} - \frac{4}{125} a^{6} + \frac{36}{125} a^{4} - \frac{61}{125} a^{3} - \frac{33}{125} a^{2} - \frac{4}{25} a + \frac{9}{25}$, $\frac{1}{625} a^{12} + \frac{6}{625} a^{10} + \frac{4}{625} a^{9} + \frac{12}{625} a^{8} - \frac{54}{625} a^{7} + \frac{6}{125} a^{6} + \frac{11}{625} a^{5} - \frac{151}{625} a^{4} + \frac{302}{625} a^{3} + \frac{62}{125} a^{2} + \frac{49}{125} a + \frac{2}{25}$, $\frac{1}{2500000} a^{13} - \frac{31}{125000} a^{12} - \frac{7159}{2500000} a^{11} + \frac{5423}{312500} a^{10} - \frac{1244}{78125} a^{9} - \frac{23827}{1250000} a^{8} - \frac{23519}{500000} a^{7} - \frac{52901}{625000} a^{6} - \frac{25837}{312500} a^{5} + \frac{326691}{1250000} a^{4} + \frac{79597}{500000} a^{3} + \frac{49067}{125000} a^{2} + \frac{31653}{100000} a - \frac{4661}{50000}$, $\frac{1}{80000000} a^{14} + \frac{1}{8000000} a^{13} + \frac{10241}{80000000} a^{12} + \frac{76607}{40000000} a^{11} + \frac{33757}{5000000} a^{10} - \frac{787347}{40000000} a^{9} + \frac{207277}{16000000} a^{8} - \frac{344227}{40000000} a^{7} - \frac{92413}{2500000} a^{6} - \frac{2788549}{40000000} a^{5} + \frac{2368129}{16000000} a^{4} + \frac{3870789}{8000000} a^{3} + \frac{1314221}{3200000} a^{2} + \frac{378617}{800000} a - \frac{36043}{160000}$

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:  $11$
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:  \( 1801375611110000000 \) (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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R R R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{3}^{2}$
2.12.22.38$x^{12} - 12 x^{8} - 8 x^{6} + 4 x^{4} + 16 x^{2} - 4$$6$$2$$22$12T50$[2, 8/3, 8/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$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$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
37Data not computed
74483599Data not computed