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* = *p*^{d} 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

*y*² = *x*³ + 4*x* + 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.