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Properties

Label 15.15.2280727624...7600.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{21}\cdot 3^{20}\cdot 5^{2}\cdot 7^{6}\cdot 13^{9}$
Root discriminant $143.62$
Ramified primes $2, 3, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T64

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2611280, 10795980, 12068280, 651653, -5916642, -2104767, 1048048, 553755, -86688, -64013, 3408, 3717, -54, -102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 102*x^13 - 54*x^12 + 3717*x^11 + 3408*x^10 - 64013*x^9 - 86688*x^8 + 553755*x^7 + 1048048*x^6 - 2104767*x^5 - 5916642*x^4 + 651653*x^3 + 12068280*x^2 + 10795980*x + 2611280)
 
gp: K = bnfinit(x^15 - 102*x^13 - 54*x^12 + 3717*x^11 + 3408*x^10 - 64013*x^9 - 86688*x^8 + 553755*x^7 + 1048048*x^6 - 2104767*x^5 - 5916642*x^4 + 651653*x^3 + 12068280*x^2 + 10795980*x + 2611280, 1)
 

Normalized defining polynomial

\( x^{15} - 102 x^{13} - 54 x^{12} + 3717 x^{11} + 3408 x^{10} - 64013 x^{9} - 86688 x^{8} + 553755 x^{7} + 1048048 x^{6} - 2104767 x^{5} - 5916642 x^{4} + 651653 x^{3} + 12068280 x^{2} + 10795980 x + 2611280 \)

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:  \(228072762463409373379710261657600=2^{21}\cdot 3^{20}\cdot 5^{2}\cdot 7^{6}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.62$
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:  $2, 3, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4573632436147151986645374861752} a^{14} - \frac{265277158274013455941891798659}{4573632436147151986645374861752} a^{13} + \frac{90858290343697231788032261651}{4573632436147151986645374861752} a^{12} + \frac{966944034387220602056484175717}{4573632436147151986645374861752} a^{11} - \frac{185483974254210106657391987097}{2286816218073575993322687430876} a^{10} + \frac{1021987988897763123544980511605}{2286816218073575993322687430876} a^{9} + \frac{429309537734918300837704051733}{4573632436147151986645374861752} a^{8} + \frac{761643494843848301841040308769}{4573632436147151986645374861752} a^{7} + \frac{110037773376943457200353342704}{571704054518393998330671857719} a^{6} + \frac{550222511004745299340423875953}{1143408109036787996661343715438} a^{5} - \frac{1942902922196288787288737408323}{4573632436147151986645374861752} a^{4} + \frac{391785376675441053701795612055}{4573632436147151986645374861752} a^{3} - \frac{4366668801059688870492016671}{10786868953177245251522110523} a^{2} + \frac{246745692559333952504281254233}{1143408109036787996661343715438} a - \frac{239913487544414083256944835789}{571704054518393998330671857719}$

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:  \( 2087659961140 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T64:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 9720
The 45 conjugacy class representatives for [3^5:2]F(5)
Character table for [3^5:2]F(5) is not computed

Intermediate fields

5.5.6889792.1

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 R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$