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Properties

Label 15.15.1886139692...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{22}\cdot 5^{7}\cdot 23^{2}\cdot 31^{2}\cdot 73^{6}$
Root discriminant $141.81$
Ramified primes $3, 5, 23, 31, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T76

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-33389427, 22949361, 30020616, -27918981, -4633212, 10239144, -1619812, -1257327, 419754, 39915, -31470, 1788, 828, -99, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 99*x^13 + 828*x^12 + 1788*x^11 - 31470*x^10 + 39915*x^9 + 419754*x^8 - 1257327*x^7 - 1619812*x^6 + 10239144*x^5 - 4633212*x^4 - 27918981*x^3 + 30020616*x^2 + 22949361*x - 33389427)
 
gp: K = bnfinit(x^15 - 6*x^14 - 99*x^13 + 828*x^12 + 1788*x^11 - 31470*x^10 + 39915*x^9 + 419754*x^8 - 1257327*x^7 - 1619812*x^6 + 10239144*x^5 - 4633212*x^4 - 27918981*x^3 + 30020616*x^2 + 22949361*x - 33389427, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 99 x^{13} + 828 x^{12} + 1788 x^{11} - 31470 x^{10} + 39915 x^{9} + 419754 x^{8} - 1257327 x^{7} - 1619812 x^{6} + 10239144 x^{5} - 4633212 x^{4} - 27918981 x^{3} + 30020616 x^{2} + 22949361 x - 33389427 \)

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:  \(188613969258415859047907606953125=3^{22}\cdot 5^{7}\cdot 23^{2}\cdot 31^{2}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $141.81$
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:  $3, 5, 23, 31, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{27557941845190485219821311802833} a^{14} - \frac{12685604713100801537085679494327}{27557941845190485219821311802833} a^{13} + \frac{6102873051817237488279159434927}{27557941845190485219821311802833} a^{12} - \frac{13290746399836163799771084311155}{27557941845190485219821311802833} a^{11} - \frac{236063953616928240307047342823}{27557941845190485219821311802833} a^{10} - \frac{6413759226945103736685768418034}{27557941845190485219821311802833} a^{9} - \frac{8041666511503518778312137984051}{27557941845190485219821311802833} a^{8} - \frac{9207382678486614946375479742934}{27557941845190485219821311802833} a^{7} - \frac{1885052735449917593021479481048}{27557941845190485219821311802833} a^{6} - \frac{2141815958777946775327377798399}{27557941845190485219821311802833} a^{5} + \frac{8070934813071190076883860214282}{27557941845190485219821311802833} a^{4} + \frac{224542968749571485087626214542}{744809239059202303238413832509} a^{3} + \frac{11838427043491541280177118382649}{27557941845190485219821311802833} a^{2} + \frac{12993934989536617104998457442317}{27557941845190485219821311802833} a + \frac{10127961125073241426122069514068}{27557941845190485219821311802833}$

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:  \( 652004532602 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T76:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29160
The 48 conjugacy class representatives for [3^5:2]A(5)
Character table for [3^5:2]A(5) is not computed

Intermediate fields

5.5.10791225.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.6.9.16$x^{6} + 3 x^{4} + 6 x^{3} + 3$$6$$1$$9$$S_3^2$$[3/2, 2]_{2}^{2}$
3.9.13.2$x^{9} + 6 x^{5} + 3 x^{3} + 3$$9$$1$$13$$C_3^2 : D_{6} $$[3/2, 3/2, 5/3]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$23$23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
31Data not computed
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$