Elkies proved *The existence of infinitely many supersingular primes for every elliptic curve over Q*. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how about the others?



The existence of infinitely many supersingular primes for every elliptic curve over Q
It's the *auxiliary prime* $l$ that must be $3 \bmod 4$;
the *supersingular* prime $p$ is not guaranteed to be congruent to $3 \bmod 4$,
and indeed the residue of $p \bmod 4$ is unpredictable
(unless the curve has CM by an order in ${\bf Q}(i)$).
Several examples on page 565 have $p \equiv 1 \bmod 4$.
For example, the curve $X\_0(11)$: $y^2+y=x^3-x^2$ has $j = -2^{12}/11$
and bad reduction only at $11$, so we can take $l=7$ because 
the Legendre symbol $(11/7)$ is $+1$. The polynomial $P\_l(X)$ is then
$P\_7(X) = X + 15^3$, and we find $P\_7(j) = 33029/11$, with 
$(33029/7) = -1$. Hence some prime factor of $33029$ is supersingular.
It so happens that $33029$ is itself prime, and congruent to $1 \bmod 4$.


