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Properties

Label 15.3.59672896351...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 5^{15}\cdot 7^{10}\cdot 13^{2}$
Root discriminant $44.85$
Ramified primes $2, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T59

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104, 800, -650, 925, -1300, 535, -910, 450, -260, 275, -26, 90, 0, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 15*x^13 + 90*x^11 - 26*x^10 + 275*x^9 - 260*x^8 + 450*x^7 - 910*x^6 + 535*x^5 - 1300*x^4 + 925*x^3 - 650*x^2 + 800*x + 104)
 
gp: K = bnfinit(x^15 + 15*x^13 + 90*x^11 - 26*x^10 + 275*x^9 - 260*x^8 + 450*x^7 - 910*x^6 + 535*x^5 - 1300*x^4 + 925*x^3 - 650*x^2 + 800*x + 104, 1)
 

Normalized defining polynomial

\( x^{15} + 15 x^{13} + 90 x^{11} - 26 x^{10} + 275 x^{9} - 260 x^{8} + 450 x^{7} - 910 x^{6} + 535 x^{5} - 1300 x^{4} + 925 x^{3} - 650 x^{2} + 800 x + 104 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5967289635125000000000000=2^{12}\cdot 5^{15}\cdot 7^{10}\cdot 13^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.85$
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:  $2, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{10} - \frac{1}{7} a^{8} - \frac{1}{4} a^{6} + \frac{3}{14} a^{5} + \frac{2}{7} a^{4} + \frac{1}{14} a^{3} - \frac{3}{28} a^{2} + \frac{1}{14} a - \frac{3}{7}$, $\frac{1}{28} a^{11} - \frac{1}{7} a^{9} - \frac{1}{4} a^{7} + \frac{3}{14} a^{6} - \frac{3}{14} a^{5} + \frac{1}{14} a^{4} + \frac{11}{28} a^{3} + \frac{1}{14} a^{2} + \frac{1}{14} a$, $\frac{1}{28} a^{12} + \frac{5}{28} a^{8} + \frac{3}{14} a^{7} - \frac{3}{14} a^{6} - \frac{1}{14} a^{5} - \frac{13}{28} a^{4} + \frac{5}{14} a^{3} - \frac{5}{14} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{364} a^{13} + \frac{5}{364} a^{12} - \frac{5}{364} a^{10} - \frac{5}{28} a^{9} - \frac{47}{364} a^{8} + \frac{33}{182} a^{7} - \frac{3}{28} a^{6} + \frac{59}{364} a^{5} + \frac{87}{364} a^{4} - \frac{3}{91} a^{3} + \frac{113}{364} a^{2} - \frac{8}{91} a + \frac{3}{7}$, $\frac{1}{364} a^{14} + \frac{1}{364} a^{12} - \frac{5}{364} a^{11} - \frac{1}{364} a^{10} - \frac{43}{182} a^{9} - \frac{89}{364} a^{8} - \frac{31}{364} a^{7} + \frac{1}{52} a^{6} - \frac{1}{14} a^{5} + \frac{73}{364} a^{4} + \frac{21}{52} a^{3} - \frac{16}{91} a^{2} + \frac{2}{13} a + \frac{1}{7}$

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:  $8$
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:  \( 2722092.8293 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T59:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 28 conjugacy class representatives for [D(5)^3:2]3
Character table for [D(5)^3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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$$[\ ]^{3}$
2.12.12.17$x^{12} + 22 x^{10} + 75 x^{8} - 12 x^{6} - 89 x^{4} + 54 x^{2} - 115$$2$$6$$12$12T29$[2, 2]^{12}$
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$