Is there a largest prime p such that J_0(p) completely splits into elliptic curves The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny factors of $J(C)$ are elliptic curves.
Now since over $\mathbb Q$ the isogeny factors of $J_0(p)$ correspond to galois orbits of newforms it seems to me that the question wether there is a largest prime $p$ such that $J_0(p)$ splits completely into elliptic curves over $\mathbb Q$ should be easier and I wonder wether the answer is already known.
I suspect that the answer is yes and that $p = 37$ is the largest prime such that $J_0(p)$ completely splits into elliptic curves. Indeed using Cremona's database of elliptic curves I verified that 37 is the biggest prime below 300000 such that $J_0(p)$ completely splits into elliptic curves.
The reason I only ask it for prime levels $p$ is that I already showed that if $p = 37$ is indeed the largest prime such that $J_0(p)$ completely splits into elliptic curves, then $N = 1200$ is the largest composite number such that $J_0(N)$ completely splits into elliptic curves.
 A: The answer, due to Jean-Pierre Serre, can be found in an unpublished note of Henri Cohen where he characterizes the odd integers $N$ such that $J_0(N)$ is isogenous to a product of elliptic curves.
For your question, $N$ is prime, and only $p=11$, $13$, $17$, $19$ et $37$ satisfy this condition. (For $p=13$, $J_0(p)$ has dimension $0$.) 
In the general case, Cohen gives the following Theorem:
Theorem. The only odd values of $N$ for which $J_0(N)$ is isogenous to a product of elliptic curves are $N\leq 21$, as well as $ N = 25$, $27$, $33$, $37$, $45$, $49$, $57$, $75$, $99$ and $121$. (When $N\leq 9$, $N=13$ and $N=25$, $J_0(N)$ is zero.)
A: In fact we know the list of all $N$, whether even or odd, for which
$J_0(N)$ is isogenous to a product of elliptic curves.  See

Takuya Yamauchi, On $\mathbb Q$-simple factors of Jacobian varieties of modular curves, Yokohama Math. J. 53 (2007), no. 2, 149-160.

An alternative proof (which also corrects a minor error in Yamauchi's list) is given in section 5 of the recent paper

Noam D. Elkies, Everett W. Howe, and Christophe Ritzenthaler: Genus bounds for curves with fixed Frobenius eigenvalues, Proc. Amer. Math. Soc. 142 (2014), 71-84.  arXiv: 1006.0822.

See page 82 of 
the online version
or page 12 of the arXiv preprint.
