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).

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