For an elliptic curve 1600072X&_mathId=si1.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=c4f369ab71e24eb876269342b932aa75" title="Click to view the MathML source">E/Q, we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p is 1600072X&_mathId=si2.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=6d796d49e9a86ed8bbaae660dca249fb">1600072X-si2.gif">, i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L -functions, we prove that if 1600072X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=972058d844e84a7dc0b545d8a6a58ea7" title="Click to view the MathML source">End(E)=OK, where K is an imaginary quadratic field of discriminant 1600072X&_mathId=si4.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=4c4849e726906cb52f630344f404b5d5" title="Click to view the MathML source">≠−3,−4, then the number of extremal primes ≤X for E is asymptotic to 1600072X&_mathId=si5.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=d4b3fb7cf0aaedd57e609c8b60e65165" title="Click to view the MathML source">X3/4/logX. We give heuristics for related conjectures.