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Properties

Label 15.9.98150665663...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{18}\cdot 3^{23}\cdot 5^{15}\cdot 19^{4}$
Root discriminant $135.77$
Ramified primes $2, 3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T95

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![764864, 798000, -2205900, -3783375, -775200, 1707168, 1026160, -61170, -184260, -29950, 10236, 2865, -180, -90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 90*x^13 - 180*x^12 + 2865*x^11 + 10236*x^10 - 29950*x^9 - 184260*x^8 - 61170*x^7 + 1026160*x^6 + 1707168*x^5 - 775200*x^4 - 3783375*x^3 - 2205900*x^2 + 798000*x + 764864)
 
gp: K = bnfinit(x^15 - 90*x^13 - 180*x^12 + 2865*x^11 + 10236*x^10 - 29950*x^9 - 184260*x^8 - 61170*x^7 + 1026160*x^6 + 1707168*x^5 - 775200*x^4 - 3783375*x^3 - 2205900*x^2 + 798000*x + 764864, 1)
 

Normalized defining polynomial

\( x^{15} - 90 x^{13} - 180 x^{12} + 2865 x^{11} + 10236 x^{10} - 29950 x^{9} - 184260 x^{8} - 61170 x^{7} + 1026160 x^{6} + 1707168 x^{5} - 775200 x^{4} - 3783375 x^{3} - 2205900 x^{2} + 798000 x + 764864 \)

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:  \(-98150665663307736000000000000000=-\,2^{18}\cdot 3^{23}\cdot 5^{15}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.77$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{13} + \frac{1}{16} a^{12} - \frac{5}{64} a^{11} - \frac{1}{32} a^{10} - \frac{15}{128} a^{9} + \frac{1}{32} a^{8} - \frac{7}{64} a^{7} + \frac{3}{32} a^{6} + \frac{23}{64} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{49}{128} a + \frac{15}{32}$, $\frac{1}{60710884279120082667283552845824} a^{14} + \frac{48628090082631610972837569921}{15177721069780020666820888211456} a^{13} - \frac{708458228641707694112674398245}{30355442139560041333641776422912} a^{12} + \frac{774848069398514376563446611081}{15177721069780020666820888211456} a^{11} - \frac{3930557069846967264133793428543}{60710884279120082667283552845824} a^{10} - \frac{17842798465975929446952613827}{237151891715312822919076378304} a^{9} + \frac{1811962179107052421013234120577}{30355442139560041333641776422912} a^{8} - \frac{437074478523947918228331480559}{15177721069780020666820888211456} a^{7} - \frac{12614250163732267437956975502065}{30355442139560041333641776422912} a^{6} - \frac{1370761202987795884067403499363}{7588860534890010333410444105728} a^{5} - \frac{372727812312675415868835081983}{948607566861251291676305513216} a^{4} + \frac{828021097498317984468391294567}{1897215133722502583352611026432} a^{3} + \frac{7400539424531357107712046012593}{60710884279120082667283552845824} a^{2} + \frac{2463032953760290144029418916863}{7588860534890010333410444105728} a + \frac{1749210697131439442643612366801}{3794430267445005166705222052864}$

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:  \( 84644812557.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T95:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 53 conjugacy class representatives for [A(5)^3:2]3 are not computed
Character table for [A(5)^3:2]3 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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 sibling: 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 R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.61$x^{12} - 6 x^{10} + 2 x^{8} - 4 x^{7} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.12.19.43$x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{6} + 3 x^{3} + 3$$12$$1$$19$$D_4 \times C_3$$[2]_{4}^{2}$
5Data not computed
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$