Computational Evidence for the Conjecture

Using a MAGMA program (see [4]), the author's computer verified Conjecture 4.1 for every $p<50$ for the first $20$ optimal elliptic curve quotients of $R_0(N)$ of rank $1$ and the first $2$ elliptic curve quotients of rank $2$.

Table 1 contains, for each $p<50$, the smallest prime $\ell$ satisfying the conditions of Conjecture 4.1. The elliptic curves are labeled as in Cremona. The curves 389A and 433A both have rank $2$, and all others have rank $1$. A dash (-) in the table indicates that the corresponding prime is not rigid, so the conjecture does not apply.

In all cases the first prime $\ell\nmid N_E$ with $\ell\equiv 1\pmod{p}$ with $a_\ell(E) \not\equiv 2\pmod{p}$ satisfied $L(E,\chi_{p,\ell},1)\neq 0$, except for 61A with $p=5$, 79A with $p=7$, 82A with $p=5$, 89A with $p=11$, and 92B with $p=5$. In every one of these $5$ exceptional cases, the second prime that we tried satisfied the conclusion of Conjecture 4.1.