Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 2 / a / Solution

For each $x\in {\mathbb{F}}_p$, count the roots of $y^2=x^3+x^2+x+1$, and add the identity at infinity. A nonzero quadratic residue gives two roots, zero gives one, and a nonsquare gives none. Equivalently, with the Legendre symbol extended by ${\chi }_p(0)=0$,The numbers of affine $y$ values, in increasing $x$ order starting at zero, areso the answer isAll four reductions are smooth algebraic curves: the original PDF gives the elliptic-curve discriminant $-2^8$, so these odd primes are primes of good reduction.