We show that, for most of the elliptic curves $\E$ over a prime finite field $\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number field containing the endomorphism ring of $\E$ over $\F_p$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over $\F_p$.

MSC Classifications:

11G20, 11N32, 11R11 show english descriptions Curves over finite and local fields [See also 14H25]
Primes represented by polynomials; other multiplicative structure of polynomial values
Quadratic extensions 11G20 - Curves over finite and local fields [See also 14H25]
11N32 - Primes represented by polynomials; other multiplicative structure of polynomial values
11R11 - Quadratic extensions

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