An example of bad reduction mod p

Elliptic curves are a natural construction over the complex numbers, but curves over the complex numbers aren’t very useful in computing. For that, we need elliptic curves that are defined over a finite field. It turns out that an elliptic curve over the integers can be reduced to one over a finite field in most cases. In particular, if we have an elliptic curve defined by

y² = x³ + ax + b

which has discriminant

D = -4a³ - 27 b²

then we want to look at what happens if reduce everything modulo a prime p. As long as p isn’t a factor of D, everything works fine. If p is a factor of D, however, then we get a singular curve when we reduce mod p. For example, the curve

y² = (x - 3)(x - 8)(x + 11)

= x³ – 97x + 264

has no repeated roots over the complex numbers, which is reflected in its non-zero discriminant

D = 1768900

= 2² 5² 7² 19²

But because 19 is a factor of D, this curve becomes singular when we reduce everything modulo 19. In particular, we find that this curve becomes

y² = x³ – 97x + 264

≡ x³ + 17x + 17 (mod 19)

 ≡ (x + 11)² (x + 16) (mod 19)

The cubic part of the curve has multiple roots so it's singular over GF(19). Modulo 19 this curve also has discriminant

D = -4(17)² – 27(17)³

= -27455

≡ 0 (mod 19)

which also tells us that this curve is singular over GF(19).