In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings.
Elliptic curves over such fields which are not supersingular are called ordinary and these two classes of elliptic curves behave fundamentally differently in many aspects.
Hasse (1936) discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and Deuring (1941) developed their basic theory.
In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular.
There are many different but equivalent ways of defining supersingular elliptic curves that have been used.
Over an algebraically closed field K an elliptic curve is determined by its j-invariant, so there are only a finite number of supersingular elliptic curves.
There are exactly ⌊p/12⌋ supersingular elliptic curves with automorphism groups of order 2.
Birch & Kuyk (1975) give a table of all j-invariants of supersingular curves for primes up to 307.
This is why it is the first with a supersingular elliptic curve whose j-invariant is not a (rational) integer.