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I guess a naive workaround would be this:
var('x,y')
solve_mod(y^2==x^3+1,10)
Result seems correct:
[(0, 1), (0, 9), (2, 7), (2, 3), (4, 5), (5, 6), (5, 4), (7, 2), (7, 8), (9, 0)]
But this has a very boring algorithm that has nothing to do with elliptic curves and surely is very slow and inefficient. Not being an expert, I would wonder whether there is a reason why this is not implemented in the first place - perhaps such "curves" over non-prime moduli aren't really well understood or do not obey the usual theorems?