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Properties

Label 15.15.9308995761...3648.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{16}\cdot 3^{16}\cdot 53^{9}$
Root discriminant $73.21$
Ramified primes $2, 3, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T56

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![311392, 965808, 744192, -488292, -870528, -146880, 254000, 95184, -31104, -17044, 1728, 1440, -36, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 - 36*x^12 + 1440*x^11 + 1728*x^10 - 17044*x^9 - 31104*x^8 + 95184*x^7 + 254000*x^6 - 146880*x^5 - 870528*x^4 - 488292*x^3 + 744192*x^2 + 965808*x + 311392)
 
gp: K = bnfinit(x^15 - 60*x^13 - 36*x^12 + 1440*x^11 + 1728*x^10 - 17044*x^9 - 31104*x^8 + 95184*x^7 + 254000*x^6 - 146880*x^5 - 870528*x^4 - 488292*x^3 + 744192*x^2 + 965808*x + 311392, 1)
 

Normalized defining polynomial

\( x^{15} - 60 x^{13} - 36 x^{12} + 1440 x^{11} + 1728 x^{10} - 17044 x^{9} - 31104 x^{8} + 95184 x^{7} + 254000 x^{6} - 146880 x^{5} - 870528 x^{4} - 488292 x^{3} + 744192 x^{2} + 965808 x + 311392 \)

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:  \(9308995761095593587893403648=2^{16}\cdot 3^{16}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.21$
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:  $2, 3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{6} + \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{36} a^{8} - \frac{1}{36} a^{7} - \frac{1}{18} a^{6} - \frac{1}{18} a^{5} + \frac{1}{18} a^{4} + \frac{1}{9} a^{3} + \frac{5}{18} a^{2} - \frac{5}{18} a + \frac{4}{9}$, $\frac{1}{36} a^{9} + \frac{1}{18} a^{6} - \frac{5}{18} a^{3} + \frac{4}{9}$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{7} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{7}{18} a + \frac{1}{3}$, $\frac{1}{36} a^{11} - \frac{1}{36} a^{7} - \frac{1}{18} a^{6} - \frac{1}{6} a^{5} + \frac{1}{18} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{5}{18} a + \frac{1}{9}$, $\frac{1}{216} a^{12} - \frac{1}{108} a^{9} + \frac{1}{36} a^{7} + \frac{1}{36} a^{6} - \frac{1}{18} a^{4} - \frac{8}{27} a^{3} + \frac{5}{18} a + \frac{4}{27}$, $\frac{1}{216} a^{13} - \frac{1}{108} a^{10} - \frac{1}{36} a^{7} + \frac{1}{18} a^{6} + \frac{4}{27} a^{4} - \frac{1}{9} a^{3} + \frac{7}{27} a - \frac{4}{9}$, $\frac{1}{17712} a^{14} + \frac{1}{8856} a^{13} + \frac{13}{8856} a^{12} + \frac{1}{1107} a^{11} - \frac{1}{4428} a^{10} + \frac{5}{1107} a^{9} - \frac{13}{1476} a^{8} + \frac{1}{246} a^{7} + \frac{113}{1476} a^{6} + \frac{109}{2214} a^{5} + \frac{185}{2214} a^{4} - \frac{247}{2214} a^{3} + \frac{431}{4428} a^{2} - \frac{25}{2214} a - \frac{383}{1107}$

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:  \( 1919192855.56 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T56:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4860
The 63 conjugacy class representatives for [3^5]F(5)=3wrF(5) are not computed
Character table for [3^5]F(5)=3wrF(5) is not computed

Intermediate fields

5.5.2382032.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ $15$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ R $15$

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$$[\ ]^{3}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.16.12$x^{12} + 132 x^{11} - 135 x^{10} + 264 x^{9} - 108 x^{8} - 81 x^{7} - 99 x^{6} + 162 x^{5} + 243 x^{4} - 135 x^{3} - 162 x^{2} + 324$$3$$4$$16$12T73$[2, 2, 2]^{4}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.9.2$x^{12} - 106 x^{8} + 2809 x^{4} - 9528128$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$