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Properties

Label 15.9.77356075654...8384.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{10}\cdot 3^{21}\cdot 137^{2}\cdot 1567^{3}$
Root discriminant $62.02$
Ramified primes $2, 3, 137, 1567$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T83

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4384, 6576, -29592, -63600, -52062, -2367, 25096, 6819, -954, 1791, 54, -90, -40, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384)
 
gp: K = bnfinit(x^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384, 1)
 

Normalized defining polynomial

\( x^{15} - 33 x^{13} - 40 x^{12} - 90 x^{11} + 54 x^{10} + 1791 x^{9} - 954 x^{8} + 6819 x^{7} + 25096 x^{6} - 2367 x^{5} - 52062 x^{4} - 63600 x^{3} - 29592 x^{2} + 6576 x + 4384 \)

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:  \(-773560756544273759852688384=-\,2^{10}\cdot 3^{21}\cdot 137^{2}\cdot 1567^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.02$
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:  $2, 3, 137, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{5}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{4} - \frac{1}{3} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a$, $\frac{1}{72} a^{13} - \frac{1}{36} a^{12} - \frac{1}{72} a^{11} + \frac{1}{36} a^{10} - \frac{1}{36} a^{9} + \frac{1}{4} a^{8} + \frac{19}{72} a^{7} + \frac{2}{9} a^{6} - \frac{25}{72} a^{5} - \frac{7}{36} a^{4} + \frac{11}{24} a^{3} + \frac{2}{9} a^{2} + \frac{7}{18} a - \frac{2}{9}$, $\frac{1}{262139234794749959801930791056} a^{14} - \frac{39482676838014513978424927}{65534808698687489950482697764} a^{13} + \frac{198226211010341098292542405}{87379744931583319933976930352} a^{12} - \frac{2580410142370500761985185117}{65534808698687489950482697764} a^{11} + \frac{5273967300377620338613836137}{43689872465791659966988465176} a^{10} - \frac{11039558336941750196938935997}{131069617397374979900965395528} a^{9} + \frac{28887552077948516425094465095}{262139234794749959801930791056} a^{8} - \frac{48525402150269316040173130367}{131069617397374979900965395528} a^{7} - \frac{37232217208283327006921706487}{87379744931583319933976930352} a^{6} - \frac{8966745323503616744197525039}{21844936232895829983494232588} a^{5} + \frac{52226391759706110606014921185}{262139234794749959801930791056} a^{4} + \frac{11042686102635054039894417023}{131069617397374979900965395528} a^{3} - \frac{13050344517034063630566490919}{32767404349343744975241348882} a^{2} - \frac{51180401335192150059325490}{1820411352741319165291186049} a - \frac{348855579838829196680513413}{16383702174671872487620674441}$

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:  \( 197659874.612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T83:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 58320
The 72 conjugacy class representatives for [3^5:2]S(5) are not computed
Character table for [3^5:2]S(5) is not computed

Intermediate fields

5.3.14103.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$$[\ ]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.12.18.49$x^{12} + 3 x^{11} - 6 x^{10} - 3 x^{9} + 9 x^{7} + 3 x^{6} - 9 x^{5} - 9 x^{4} - 9$$6$$2$$18$12T119$[3/2, 2, 2]_{2}^{4}$
137Data not computed
1567Data not computed