# Is there an elliptic curve with 2024 points?

On New Year’s Day I posted about groups of order 2024. Are there elliptic curves of order 2024?

The Hasse-Weil theorem relates the number of points on an elliptic curve over a finite field to the number of elements of the field. Namely, an elliptic curve E over a field with q elements must have cardinality

q + 1 − t

where

|t| ≤ 2√q.

So if there is an elliptic curve with 2024 points, the curve must be over a field with roughly 2024 points.

The condition on t above is necessary for the existence of an elliptic curve of a certain size, but is it sufficient? Sorta.

The order of a finite field must be a prime power, i.e. q = pd for some prime p. There is a theorem ([1], Theorem 13.30) that there exists a curve of the size indicated in the Hasse-Weil theorem if t ≠ 0 mod p. The theorem also lists a couple more sufficient conditions that are more complicated.

So, for example, we could take q = p = 2027 and t = 4.

Now that we know the search isn’t futile, we can search for an elliptic curve over the integers mod 2027 that has 2024 points. After a brief brute force search I found

y² = x³ + 4x + 28

over the field with 2027 elements is such a curve .

## Related posts

[1] Henri Coghen and Gerhard Frey. Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC. 2006.