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Properties

Label 15.1.43519916024...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 5^{15}\cdot 13^{3}\cdot 19^{5}$
Root discriminant $51.20$
Ramified primes $2, 5, 13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-424, -2820, -6930, -4495, 950, 4438, 180, -625, -1220, 400, -226, 255, -10, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 30*x^13 - 10*x^12 + 255*x^11 - 226*x^10 + 400*x^9 - 1220*x^8 - 625*x^7 + 180*x^6 + 4438*x^5 + 950*x^4 - 4495*x^3 - 6930*x^2 - 2820*x - 424)
 
gp: K = bnfinit(x^15 + 30*x^13 - 10*x^12 + 255*x^11 - 226*x^10 + 400*x^9 - 1220*x^8 - 625*x^7 + 180*x^6 + 4438*x^5 + 950*x^4 - 4495*x^3 - 6930*x^2 - 2820*x - 424, 1)
 

Normalized defining polynomial

\( x^{15} + 30 x^{13} - 10 x^{12} + 255 x^{11} - 226 x^{10} + 400 x^{9} - 1220 x^{8} - 625 x^{7} + 180 x^{6} + 4438 x^{5} + 950 x^{4} - 4495 x^{3} - 6930 x^{2} - 2820 x - 424 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-43519916024000000000000000=-\,2^{18}\cdot 5^{15}\cdot 13^{3}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.20$
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:  $2, 5, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} + \frac{7}{32} a^{4} + \frac{5}{32} a^{3} + \frac{7}{16} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} - \frac{1}{32} a^{7} - \frac{1}{8} a^{6} - \frac{3}{64} a^{5} - \frac{1}{16} a^{4} + \frac{3}{64} a^{3} - \frac{11}{32} a^{2} + \frac{1}{16} a - \frac{3}{8}$, $\frac{1}{137741740973454208} a^{14} - \frac{854384787375925}{137741740973454208} a^{13} + \frac{1715178167025887}{137741740973454208} a^{12} + \frac{2800785083631627}{137741740973454208} a^{11} + \frac{266734857478367}{8608858810840888} a^{10} + \frac{2413489994669719}{68870870486727104} a^{9} - \frac{2718829743680251}{68870870486727104} a^{8} - \frac{3979285738926915}{68870870486727104} a^{7} + \frac{3827464906002317}{137741740973454208} a^{6} + \frac{31275959807552915}{137741740973454208} a^{5} - \frac{13092318969969537}{137741740973454208} a^{4} - \frac{16980693670042533}{137741740973454208} a^{3} + \frac{1688523477515249}{68870870486727104} a^{2} + \frac{8073333584883109}{34435435243363552} a - \frac{4252625972762265}{17217717621681776}$

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:  $7$
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:  \( 33602302.7316 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.1.76.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R $15$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
5Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$