Describe the integral solutions to $y^2 = 12x^3 - 39$

Question

Does the above Diophantine equation have infinitely many integer solutions ? One such solution is $(x,y) = (4,27)$.

Answer 1

Multiply both sides by $144,$ you get a Mordell curve $U^2 = V^3 - 5616.$

This has only the solutions you knew about

E_-05616: r = 1   t = 1   #III =  1
          E(Q) = <(48, 324)>
          R =   1.0595282130
           2 integral points
             1. (48, 324) = 1 * (48, 324)
             2. (48, -324) = -(48, 324)

http://tnt.math.se.tmu.ac.jp/simath/MORDELL/MORDELL-