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Properties

Label 15.5.19260812546...4512.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{6}\cdot 23^{5}\cdot 881^{6}$
Root discriminant $56.53$
Ramified primes $2, 23, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_5 \times S_3$ (as 15T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 0, -3726, 5589, 3789, -6678, -1727, 2691, 560, -1058, -74, 188, -10, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 11*x^13 - 10*x^12 + 188*x^11 - 74*x^10 - 1058*x^9 + 560*x^8 + 2691*x^7 - 1727*x^6 - 6678*x^5 + 3789*x^4 + 5589*x^3 - 3726*x^2 + 729)
 
gp: K = bnfinit(x^15 - 11*x^13 - 10*x^12 + 188*x^11 - 74*x^10 - 1058*x^9 + 560*x^8 + 2691*x^7 - 1727*x^6 - 6678*x^5 + 3789*x^4 + 5589*x^3 - 3726*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{15} - 11 x^{13} - 10 x^{12} + 188 x^{11} - 74 x^{10} - 1058 x^{9} + 560 x^{8} + 2691 x^{7} - 1727 x^{6} - 6678 x^{5} + 3789 x^{4} + 5589 x^{3} - 3726 x^{2} + 729 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-192608125464899891336664512=-\,2^{6}\cdot 23^{5}\cdot 881^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.53$
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:  $2, 23, 881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{2}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{54} a^{12} - \frac{1}{18} a^{11} - \frac{1}{27} a^{10} + \frac{23}{54} a^{9} - \frac{8}{27} a^{8} - \frac{13}{27} a^{7} + \frac{19}{54} a^{6} + \frac{17}{54} a^{5} - \frac{1}{9} a^{4} - \frac{4}{27} a^{3} + \frac{5}{18} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{162} a^{13} + \frac{7}{162} a^{11} + \frac{17}{162} a^{10} + \frac{71}{162} a^{9} + \frac{8}{81} a^{8} + \frac{31}{162} a^{7} - \frac{35}{81} a^{6} - \frac{5}{18} a^{5} - \frac{22}{81} a^{4} - \frac{7}{18} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{84528359491551341914134} a^{14} - \frac{3599366907955981573}{4696019971752852328563} a^{13} + \frac{228840905121129535706}{42264179745775670957067} a^{12} + \frac{1809501213344708608285}{42264179745775670957067} a^{11} - \frac{3908341094371050485269}{84528359491551341914134} a^{10} + \frac{6592659569480601161791}{84528359491551341914134} a^{9} + \frac{5971124110502746225885}{84528359491551341914134} a^{8} + \frac{16443525631679972940061}{42264179745775670957067} a^{7} + \frac{576687035721123252836}{4696019971752852328563} a^{6} + \frac{29187253126073215248571}{84528359491551341914134} a^{5} - \frac{170720784419672915113}{1043559993722856073014} a^{4} - \frac{1973184608090663553299}{9392039943505704657126} a^{3} + \frac{380795087839524981449}{1565339990584284109521} a^{2} - \frac{14164064994828953387}{347853331240952024338} a - \frac{91793070767891068877}{347853331240952024338}$

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:  $9$
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:  \( 16317072.209 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_5$ (as 15T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 15 conjugacy class representatives for $A_5 \times S_3$
Character table for $A_5 \times S_3$

Intermediate fields

3.1.23.1, 5.5.3104644.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$23$23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
881Data not computed