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Properties

Label 15.11.2933480855...2688.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{10}\cdot 3^{20}\cdot 13^{2}\cdot 36497^{3}$
Root discriminant $79.04$
Ramified primes $2, 3, 13, 36497$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T91

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5408, -40560, -121680, -186376, -160026, -93123, -50696, -13401, 9180, 4618, 84, 63, -22, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 33*x^13 - 22*x^12 + 63*x^11 + 84*x^10 + 4618*x^9 + 9180*x^8 - 13401*x^7 - 50696*x^6 - 93123*x^5 - 160026*x^4 - 186376*x^3 - 121680*x^2 - 40560*x - 5408)
 
gp: K = bnfinit(x^15 - 33*x^13 - 22*x^12 + 63*x^11 + 84*x^10 + 4618*x^9 + 9180*x^8 - 13401*x^7 - 50696*x^6 - 93123*x^5 - 160026*x^4 - 186376*x^3 - 121680*x^2 - 40560*x - 5408, 1)
 

Normalized defining polynomial

\( x^{15} - 33 x^{13} - 22 x^{12} + 63 x^{11} + 84 x^{10} + 4618 x^{9} + 9180 x^{8} - 13401 x^{7} - 50696 x^{6} - 93123 x^{5} - 160026 x^{4} - 186376 x^{3} - 121680 x^{2} - 40560 x - 5408 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29334808557573827860354802688=2^{10}\cdot 3^{20}\cdot 13^{2}\cdot 36497^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.04$
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:  $2, 3, 13, 36497$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{4} - \frac{2}{9} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{5} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{9} - \frac{1}{9} a^{3} - \frac{2}{27}$, $\frac{1}{27} a^{10} - \frac{1}{9} a^{4} - \frac{2}{27} a$, $\frac{1}{216} a^{11} - \frac{1}{216} a^{9} - \frac{1}{36} a^{8} - \frac{1}{24} a^{7} - \frac{1}{18} a^{6} - \frac{1}{36} a^{5} - \frac{1}{6} a^{4} + \frac{5}{72} a^{3} - \frac{10}{27} a^{2} - \frac{1}{8} a - \frac{41}{108}$, $\frac{1}{5184} a^{12} + \frac{1}{864} a^{11} - \frac{1}{576} a^{10} + \frac{13}{1296} a^{9} - \frac{5}{576} a^{8} + \frac{5}{96} a^{7} - \frac{17}{864} a^{6} - \frac{1}{24} a^{5} + \frac{29}{192} a^{4} + \frac{65}{2592} a^{3} + \frac{647}{1728} a^{2} + \frac{41}{144} a - \frac{71}{1296}$, $\frac{1}{41472} a^{13} - \frac{1}{10368} a^{12} - \frac{23}{13824} a^{11} - \frac{25}{20736} a^{10} - \frac{757}{41472} a^{9} - \frac{1}{96} a^{8} + \frac{109}{6912} a^{7} + \frac{115}{3456} a^{6} + \frac{263}{4608} a^{5} + \frac{811}{10368} a^{4} + \frac{5825}{41472} a^{3} - \frac{301}{6912} a^{2} + \frac{4543}{10368} a - \frac{1037}{5184}$, $\frac{1}{4313088} a^{14} + \frac{1}{165888} a^{13} - \frac{7}{159744} a^{12} - \frac{1033}{539136} a^{11} - \frac{37585}{4313088} a^{10} - \frac{12817}{718848} a^{9} - \frac{18947}{718848} a^{8} + \frac{4523}{179712} a^{7} - \frac{427}{159744} a^{6} + \frac{69383}{2156544} a^{5} + \frac{107753}{4313088} a^{4} + \frac{17}{29952} a^{3} - \frac{10453}{539136} a^{2} - \frac{3779}{10368} a + \frac{1723}{6912}$

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:  $12$
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:  \( 1315631457.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T91:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 72 conjugacy class representatives for [1/2.S(3)^5]S(5) are not computed
Character table for [1/2.S(3)^5]S(5) is not computed

Intermediate fields

5.5.36497.1

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
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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.19$x^{12} + 66 x^{11} - 45 x^{10} - 120 x^{9} + 9 x^{8} + 108 x^{7} + 18 x^{6} - 108 x^{5} - 81 x^{4} - 54 x^{3} - 81 x^{2} - 81$$3$$4$$16$12T46$[2, 2]^{8}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.12.0.1$x^{12} + x^{2} - x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
36497Data not computed