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Properties

Label 15.3.13129924718...3125.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 5^{9}\cdot 53^{3}\cdot 109^{3}$
Root discriminant $64.25$
Ramified primes $3, 5, 53, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T75

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5779, -14535, -12390, 6090, 17475, 8994, -3905, -4560, -1080, 25, -54, 255, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 30*x^13 + 255*x^11 - 54*x^10 + 25*x^9 - 1080*x^8 - 4560*x^7 - 3905*x^6 + 8994*x^5 + 17475*x^4 + 6090*x^3 - 12390*x^2 - 14535*x - 5779)
 
gp: K = bnfinit(x^15 + 30*x^13 + 255*x^11 - 54*x^10 + 25*x^9 - 1080*x^8 - 4560*x^7 - 3905*x^6 + 8994*x^5 + 17475*x^4 + 6090*x^3 - 12390*x^2 - 14535*x - 5779, 1)
 

Normalized defining polynomial

\( x^{15} + 30 x^{13} + 255 x^{11} - 54 x^{10} + 25 x^{9} - 1080 x^{8} - 4560 x^{7} - 3905 x^{6} + 8994 x^{5} + 17475 x^{4} + 6090 x^{3} - 12390 x^{2} - 14535 x - 5779 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1312992471874372026626953125=3^{20}\cdot 5^{9}\cdot 53^{3}\cdot 109^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.25$
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:  $3, 5, 53, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{31621351703624086147915} a^{14} - \frac{2905448010759117797718}{31621351703624086147915} a^{13} - \frac{535282950830783050621}{6324270340724817229583} a^{12} + \frac{9478477939054976674101}{31621351703624086147915} a^{11} + \frac{8343545131259369225702}{31621351703624086147915} a^{10} - \frac{2175073339634913062772}{6324270340724817229583} a^{9} - \frac{2966390554356938295001}{31621351703624086147915} a^{8} - \frac{1256377700006222180218}{6324270340724817229583} a^{7} + \frac{4144826327305431649244}{31621351703624086147915} a^{6} + \frac{3525697599546335608567}{31621351703624086147915} a^{5} + \frac{3266020134827187839301}{31621351703624086147915} a^{4} - \frac{3206270618050130656621}{31621351703624086147915} a^{3} + \frac{7331162290093813124774}{31621351703624086147915} a^{2} - \frac{3054295557059542519336}{31621351703624086147915} a + \frac{12952269198874127214596}{31621351703624086147915}$

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:  $8$
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:  \( 31912240.5834 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T75:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 55 conjugacy class representatives for [F(5)^3]3=F(5)wr3 are not computed
Character table for [F(5)^3]3=F(5)wr3 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 30 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ R ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.12.9.3$x^{12} - 25 x^{4} + 250$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
53.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.4.3.3$x^{4} + 654$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$