TITLE: Least supersingular prime
QUESTION [10 upvotes]: Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?

REPLY [6 votes]: edit I just saw that I misread the question. I leave my posting here since I hope that it is interesting for you anyway. /edit
I don't know if you are still interested in the answer but I am currently working on exactly this problem. The result I have is explicit and does not depend on any unproven conjecture. It is one chapter of my PhD thesis (which is not finished yet) and details of the proof would probably go beyond the scope of this thread.
My result ist this:
Let $E$ be an elliptic curve over the rationals with $j$-invariant $j_E$ and conductor $N$. Let \begin{align*}B_{j_E} =
\begin{cases}
\left( \frac{\log j_E}{2 \pi}\right)^2 &\mbox{for } j_E>0,\\
\left(\frac{\log |j_E|}{\pi} + 1\right)^2 &\mbox{for } j_E<0\\
\end{cases},\end{align*} $M \in\mathbb{N}$, $q = 4 \text{rad} (6 N)$ and $n= \max (3, M, B_{j_E})$. For better readability we will write $A = \frac54q(n+2\log q)^2$. Then there exists a supersingular prime $p$ of $E$ such that $p \geq n$ and
\begin{align*}\log p \leq 1.7 \cdot 10^{26} A^{13587505A} h^*(j_E).\end{align*}
The "+1" in the definition of $B_{j_E}$ can probably be omitted but it would make one part of the proof more complicated. Since it only affects the result very very slightly I kept it there.
The bound should be the same for number fields of odd degree. It may be possible to get a similar result on number fields with at least one real embedding (since Elkies proved the existance of inifinitely many supersingular primes also for elliptic curves over number fields with at least one real embedding) where the methods have to be adjusted slightly.