Score:1

Distribution of elliptic curves with rank 2?

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An elliptic curve defined over a finite field is either cyclic, or a direct sum of two cyclic groups. In cryptography, we use exclusively the former. I was wondering if there is any result on how common (or rare) are rank-2 elliptic curves defined over $\mathbb{F}_{q}$ for a prime power $q$.

(They don't seem rare. For a fixed prime $p$ of 64 bits, I sampled 1000 random curves defined over $\mathbb{F}_p$ and got 170 curves of rank 2. Same experiment with a 128-bit $p$ yields 208 such curves. )

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