Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 3 / b / Solution

The supplied integral equation has elliptic-curve discriminant $\Delta =-11$. It therefore has good reduction at every $p\ne 11$. At eleven its discriminant has valuation one. Under a change of Weierstrass equation of an elliptic curve, discriminant valuations differ by a multiple of twelve. A good integral model would have valuation zero, which is impossible. This is the discriminant valuation obstruction to good reduction; it also shows the displayed model is minimal at eleven. Hence