[radicaux] Expression brute de cos(2π/19).

1 files changed, 18 insertions, 0 deletions

diff --git a/divers/sageries/racine-19e-de-1 b/divers/sageries/racine-19e-de-1
new file mode 100644
index 0000000..b9aa12c
--- /dev/null
+++ b/divers/sageries/racine-19e-de-1
@@ -0,0 +1,18 @@
+K.<a> = CyclotomicField(342)
+omega = a^18
+zeta = a^19
+alpha = [sum([zeta^(i*j)*omega^(2^i) for i in range(18)]) for j in range(18)]
+powtab = [NN(18/gcd(i,18)) for i in range(18)]
+atab = [alpha[i]^powtab[i] for i in range(18)]
+atab_on_zeta_basis = [(QQ^6)((zeta.coordinates_in_terms_of_powers())(x)) for x in atab]
+sqrtm3 = 2*zeta^3-1
+nice_basis = [1, sqrtm3, zeta, zeta^-1, -sqrtm3*zeta^5, sqrtm3*zeta^-5]
+m = Matrix(QQ, 6, 6, [(QQ^6)((zeta.coordinates_in_terms_of_powers())(x)) for x in nice_basis])
+atab_on_nice_basis = [v * m.inverse() for v in atab_on_zeta_basis]
+zetab = [ZZ(floor(arg(CC(N(alpha[i])/N(atab[i]^(1/powtab[i]))))/arg(zeta)+0.5)) for i in range(18)]
+btab = [zeta^zetab[i] for i in range(18)]
+btab_on_zeta_basis = [(QQ^6)((zeta.coordinates_in_terms_of_powers())(x)) for x in btab]
+btab_on_nice_basis = [v * m.inverse() for v in btab_on_zeta_basis] 
+symbolic_basis = [1, sqrt(-3), ((1/2)*(1+sqrt(-3)))^(1/3), ((1/2)*(1-sqrt(-3)))^(1/3), ((1/2)*(9+3*sqrt(-3)))^(1/3), ((1/2)*(9-3*sqrt(-3)))^(1/3)]
+symbolic_omega = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)]))^(1/powtab[i]) for i in range(18)])/18
+symbolic_cos = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(6)]))^(1/powtab[i]) for i in range(0,18,2)])/18