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Cubic forms, powers of primes and classification of elliptic curves

We prove that for almost all primes p the equation (x+y)(x+Bxy+y) = pz with B ∈ {0, 1, 4, 6} has no solution in pairwise coprime nonzero integers x, y, z 6= ±1, integer α > 0 and prime n ≥ p. Proving this we classify primes p for which there exists an elliptic curve over Q with conductor 48p, 192p or 384p and with nontrivial rational 2-torsion.