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Properties

Label 15.7.52287425129...3529.1
Degree $15$
Signature $[7, 4]$
Discriminant $11^{12}\cdot 23\cdot 38281\cdot 189223$
Root discriminant $38.13$
Ramified primes $11, 23, 38281, 189223$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T81

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-131, 1604, -6069, 11749, -13758, 9908, -3448, -714, 1475, -886, 325, -39, -25, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 14*x^13 - 25*x^12 - 39*x^11 + 325*x^10 - 886*x^9 + 1475*x^8 - 714*x^7 - 3448*x^6 + 9908*x^5 - 13758*x^4 + 11749*x^3 - 6069*x^2 + 1604*x - 131)
 
gp: K = bnfinit(x^15 - 5*x^14 + 14*x^13 - 25*x^12 - 39*x^11 + 325*x^10 - 886*x^9 + 1475*x^8 - 714*x^7 - 3448*x^6 + 9908*x^5 - 13758*x^4 + 11749*x^3 - 6069*x^2 + 1604*x - 131, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 14 x^{13} - 25 x^{12} - 39 x^{11} + 325 x^{10} - 886 x^{9} + 1475 x^{8} - 714 x^{7} - 3448 x^{6} + 9908 x^{5} - 13758 x^{4} + 11749 x^{3} - 6069 x^{2} + 1604 x - 131 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(522874251292437641653529=11^{12}\cdot 23\cdot 38281\cdot 189223\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.13$
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:  $11, 23, 38281, 189223$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{5} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{3}{16}$, $\frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{16} a - \frac{1}{8}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} + \frac{3}{16} a^{2} - \frac{1}{8} a - \frac{5}{16}$, $\frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{3}{32} a^{5} + \frac{3}{64} a^{4} - \frac{5}{64} a^{3} + \frac{1}{64} a^{2} + \frac{1}{16} a + \frac{1}{64}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{8} - \frac{1}{64} a^{7} - \frac{3}{64} a^{5} + \frac{3}{32} a^{3} - \frac{13}{64} a^{2} - \frac{3}{64} a + \frac{7}{64}$, $\frac{1}{64} a^{11} + \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{7}{64} a^{3} - \frac{1}{64} a^{2} - \frac{1}{64} a - \frac{3}{32}$, $\frac{1}{256} a^{12} - \frac{1}{128} a^{10} - \frac{7}{256} a^{8} - \frac{1}{64} a^{7} + \frac{3}{128} a^{6} + \frac{3}{64} a^{5} + \frac{3}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{16} a^{2} + \frac{9}{64} a + \frac{33}{256}$, $\frac{1}{256} a^{13} - \frac{1}{128} a^{11} + \frac{1}{256} a^{9} + \frac{1}{64} a^{8} - \frac{1}{128} a^{7} - \frac{1}{64} a^{6} - \frac{5}{128} a^{5} + \frac{1}{16} a^{4} + \frac{1}{32} a^{3} - \frac{5}{64} a^{2} + \frac{1}{256} a + \frac{1}{32}$, $\frac{1}{256} a^{14} + \frac{1}{256} a^{10} + \frac{1}{64} a^{8} + \frac{1}{64} a^{7} - \frac{3}{128} a^{6} - \frac{3}{64} a^{5} + \frac{3}{32} a^{4} + \frac{1}{32} a^{3} - \frac{23}{256} a^{2} + \frac{1}{64} a + \frac{5}{128}$

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:  $10$
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:  \( 1332210.24405 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T81:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 63 conjugacy class representatives for [S(3)^5]5=S(3)wr5 are not computed
Character table for [S(3)^5]5=S(3)wr5 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 $15$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ $15$

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
11Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
38281Data not computed
189223Data not computed