The motivation for this question comes from the study of elliptic curves, but the question itself is about the choice of convenient algebraic transformations.

Fix a base ring (commutative, with a multiplicative identity) $ R $. We consider the polynomial equation

$$

y ^ 2 + a_1xy + a_3y = x ^ 3 + a_2x ^ 2 + a_4x + a_6

$$

where $ a_i $ stay in bed $ R $. The allowed coordinate transformations are

$$

x = x & # 39; + r, qquad y = y & # 39; + sx & # 39; + t, qquad r, s, t in R.

$$

It can be verified that under such a transformation, the shape of the polynomial equation is preserved but $ a_i $ change its value (in some explicit way I don't want to write at this time).

Yes, both $ 2 $ Y $ 3 $ are units in $ R $, there is a unique choice of coordinates so that $ a_1 = a_2 = a_3 = 0 $.

Yes $ 2 = 0 in R $ (resp. $ 3 = 0 in R $) there are sub-boxes, but you can choose (not exclusively) a good form for the equation and describe all the transformations that fix it (Deligne, Courbes elliptique: formulaire).

I want to consider $ R = mathbb {Z} / m mathbb {Z} $ for $ m> $ 3 (particularly $ m = $ a power of $ 2 $ or $ 3 $) Has anyone discovered what is a good way for our equation on this base ring (and what are the transformations that preserve it)?