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Properties

Label 15.15.5927791845...9216.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{20}\cdot 11^{12}\cdot 23^{2}$
Root discriminant $71.04$
Ramified primes $2, 3, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T71

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6368, 46080, 102744, 35268, -125814, -95931, 50068, 50136, -8370, -10762, 612, 1107, -16, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 54*x^13 - 16*x^12 + 1107*x^11 + 612*x^10 - 10762*x^9 - 8370*x^8 + 50136*x^7 + 50068*x^6 - 95931*x^5 - 125814*x^4 + 35268*x^3 + 102744*x^2 + 46080*x + 6368)
 
gp: K = bnfinit(x^15 - 54*x^13 - 16*x^12 + 1107*x^11 + 612*x^10 - 10762*x^9 - 8370*x^8 + 50136*x^7 + 50068*x^6 - 95931*x^5 - 125814*x^4 + 35268*x^3 + 102744*x^2 + 46080*x + 6368, 1)
 

Normalized defining polynomial

\( x^{15} - 54 x^{13} - 16 x^{12} + 1107 x^{11} + 612 x^{10} - 10762 x^{9} - 8370 x^{8} + 50136 x^{7} + 50068 x^{6} - 95931 x^{5} - 125814 x^{4} + 35268 x^{3} + 102744 x^{2} + 46080 x + 6368 \)

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:  \(5927791845298029219947529216=2^{10}\cdot 3^{20}\cdot 11^{12}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.04$
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:  $2, 3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{3}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{2923372625713616305357552} a^{14} - \frac{9098801335327574999879}{1461686312856808152678776} a^{13} + \frac{24504523209343839218199}{1461686312856808152678776} a^{12} + \frac{20691868964722228293655}{730843156428404076339388} a^{11} + \frac{1290248289144774755583291}{2923372625713616305357552} a^{10} + \frac{586611221451808997094461}{1461686312856808152678776} a^{9} + \frac{276355138768229469706789}{1461686312856808152678776} a^{8} - \frac{10409276501334073883729}{63551578819861224029512} a^{7} - \frac{35176329361857877278897}{730843156428404076339388} a^{6} - \frac{167791379252108426290289}{730843156428404076339388} a^{5} + \frac{651780132973924851599725}{2923372625713616305357552} a^{4} + \frac{178497306780674515968217}{730843156428404076339388} a^{3} - \frac{195925540212982100180417}{730843156428404076339388} a^{2} - \frac{17179239062639173136304}{182710789107101019084847} a + \frac{58352560464660728939454}{182710789107101019084847}$

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:  \( 1280430134.04 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T71:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 19440
The 39 conjugacy class representatives for [1/2.S(3)^5]5
Character table for [1/2.S(3)^5]5 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\)

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
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 $15$ $15$ R $15$ $15$ $15$ R $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
3Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$