Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 32 / 1 / i / Solution

For the Weierstrass equation of an elliptic curve in this part, $a_1=a_2=a_6=0$, $a_3=1$, $a_4=-1$. HenceThe elliptic-curve discriminant is thereforeEvery prime other than $37$ has unit discriminant, so this integral equation gives good reduction. At $37$ the valuation is one. An Admissible change of Weierstrass coordinates changes discriminant valuations by a multiple of twelve; consequently no integral model can lower this value to zero. Thus the equation is minimal there and the reduction is genuinely bad, rather than an artifact of a nonminimal model. Explicitly $(5,18)$ is singular on the reduced affine cubic: both partial derivatives vanish there. The set of bad reduction primes is