I want to determine the minimal discriminant of $$ y^2 + xy = x^3-x^2-50x+111 $$ as an elliptic curve over the rationals. I managed to reduce it to the form $y^2=x^3+Ax+B$ where $A,B$ are rational, and obtained that the discriminant is $\Delta=5^37^413$ I then transformed it to get $A,B \in \mathbb{Z}$ and my discriminant is now $2^{12} \Delta$. Following Silverman (VII.1.1) I would tend to think that this guarantees that $\Delta$ is minimal, but I cannot deduce it directly from his argument, which in addition only concerns local fields...Can anyone confirm that my $\Delta$ is indeed minimal?
Thanks !
Your original model, i.e., $y^2+xy=x^3-x^2-50x+111$, is minimal. Its discriminant is $$\Delta=3901625=5^3\cdot 7^4\cdot 13.$$ Since the coefficients of the model are integral, and the valuation of the discriminant at every prime is less than $12$, it follows that the curve is minimal at every prime number (Silverman's "The Arithmetic of Elliptic Curves", Remark 1.1 in Chapter VII).