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elliptic curve; complex multiplication field; Frobenius discriminant
We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field $\mathbb {F}_p$ of $p$ elements, such that the discriminant $D(E)$ of the quadratic number field containing the endomorphism ring of $E$ over $\mathbb {F}_p$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).
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