The Conjecture

Let $E$ be an elliptic curve over  $\mathbb{Q}$, and suppose $p$ is a rigid prime for $E$. For every prime $\ell\equiv 1\pmod{p}$, let $\chi_{p,\ell} : (\mathbb{Z}/\ell\mathbb{Z})^* \rightarrow\!\!\!\!\rightarrow \boldsymbol{\mu}_p$ be one of the Galois-conjugate characters of order $p$ and modulus $\ell$.

Conjecture 4.1   There exists a prime  $\ell\nmid N_E$ such that

$$L(E,\chi_{p,\ell},1)\neq 0$$ and $$a_\ell(E) \not\equiv 2\pmod{p}.$$

The condition $a_\ell(E) \not\equiv 2\pmod{p}$ requires elaboration. Since $\ell\equiv 1\pmod{p}$, this condition can be rewritten $a_\ell(E)\not\equiv \ell+1\pmod{p}$, which is a ``familiar'' condition to impose. We demand that $a_\ell(E)\not\equiv \ell+1\pmod{p}$ because then the characteristic polynomial $x^2 + a_\ell x +\ell\in \mathbb{F}_p[x]$ of $\Frob_\ell$ on $E[p]$ does not have $+1$ as an eigenvalue. This is a key hypothesis in Section 5.

Table 1: Evidence for Conjecture 4.1

$E$	3	5	7	11	13	17	19	23	29	31	37	41	43	47
37A	13	11	29	67	53	103	191	47	59	311	-	83	173	283
43A	7	11	29	23	53	103	191	47	59	311	149	83	-	283
53A	13	11	29	23	53	103	191	47	59	311	149	83	173	283
57A	-	11	29	23	53	103	-	47	59	311	149	83	173	283
58A	7	11	29	23	53	103	191	47	-	311	149	83	173	283
61A	7	31	29	67	53	103	191	47	59	311	149	83	173	283
65A	19	-	43	23	-	137	191	47	59	311	149	83	173	659
77A	19	11	-	-	53	103	191	47	59	311	149	83	173	283
79A	13	11	43	67	53	103	191	47	59	311	149	83	173	283
82A	13	41	29	23	53	103	191	47	59	311	149	-	173	283
83A	7	11	29	23	53	103	191	47	59	311	149	83	173	283
88A	7	11	29	-	131	103	191	47	59	311	149	83	173	283
89A	19	11	29	67	53	103	191	47	59	311	149	83	173	283
91A	31	11	-	23	-	103	191	47	59	311	149	83	173	283
91B	-	11	-	23	-	103	191	47	59	311	149	83	173	283
92B	13	61	29	23	79	103	229	-	59	311	149	83	173	283
99A	-	11	29	-	53	103	191	47	59	311	149	83	173	283
101A	7	11	29	23	53	103	191	47	59	311	149	83	173	283
102A	-	11	29	23	53	-	191	47	59	311	149	83	173	283
106B	7	11	29	23	53	137	191	47	59	311	149	83	431	283
389A	7	11	29	23	53	103	191	47	59	311	149	83	173	283
433A	7	11	43	23	53	103	191	47	59	311	149	83	173	283