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Properties

Label 15.5.28122063322...1072.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{10}\cdot 11^{12}\cdot 43\cdot 197\cdot 1033$
Root discriminant $31.38$
Ramified primes $2, 11, 43, 197, 1033$
Class number $1$
Class group Trivial
Galois group 15T81

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 0, 96, -96, -80, 160, -76, -12, 24, -52, 71, -43, 2, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 10*x^13 + 2*x^12 - 43*x^11 + 71*x^10 - 52*x^9 + 24*x^8 - 12*x^7 - 76*x^6 + 160*x^5 - 80*x^4 - 96*x^3 + 96*x^2 - 32)
 
gp: K = bnfinit(x^15 - 5*x^14 + 10*x^13 + 2*x^12 - 43*x^11 + 71*x^10 - 52*x^9 + 24*x^8 - 12*x^7 - 76*x^6 + 160*x^5 - 80*x^4 - 96*x^3 + 96*x^2 - 32, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 10 x^{13} + 2 x^{12} - 43 x^{11} + 71 x^{10} - 52 x^{9} + 24 x^{8} - 12 x^{7} - 76 x^{6} + 160 x^{5} - 80 x^{4} - 96 x^{3} + 96 x^{2} - 32 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-28122063322027324931072=-\,2^{10}\cdot 11^{12}\cdot 43\cdot 197\cdot 1033\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.38$
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:  $2, 11, 43, 197, 1033$
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 \(a\))

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

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 183908.779975 \)
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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ $15$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$[\ ]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
$197$$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
197.2.1.2$x^{2} + 394$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
197.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
1033Data not computed