Examples
- If K is a field of characteristic 2, every elliptic curve defined by an equation of the form
is supersingular (see Washington2003, p. 122).
- If K is a field of characteristic 3, every elliptic curve defined by an equation of the form
is supersingular (see Washington2003, p. 122).
- For with p>3 we have that the elliptic curve defined by is supersingular if and only if and the elliptic curve defined by is supersingular if and only if (see Washington2003, 4.35).
- There are also more exotic examples: The elliptic curve given by is nonsingular over for . It is supersingular for p = 23 and ordinary for every other (see Hartshorne1977, 4.23.6).
- Elkies (1987) showed that any elliptic curve defined over the rationals is supersingular for an infinite number of primes.
- Birch & Kuyk (1975) give a table of all supersingular curves for primes up to 307. For the first few primes the supersingular elliptic curves are given as follows. The number of supersingular values of j other than 0 or 1728 is the integer part of (p−1)/12.
| prime | supersingular j invariants |
|---|---|
| 2 | 0 |
| 3 | 0=1728 |
| 5 | 0 |
| 7 | 6=1728 |
| 11 | 0, 1=1728 |
| 13 | 5 |
| 17 | 0,8 |
| 19 | 7, 1728 |
| 23 | 0,19, 1728 |
| 29 | 0,2, 25 |
| 31 | 2, 4, 1728 |
| 37 | 8, 3±√15 |
Read more about this topic: Supersingular Elliptic Curve
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