Table of Strong Visibility at Higher Level

The following is a table that gives the known examples of ${A_f}_{/\mathbb{Q}}$ with square free conductor $N \leq 1339$ , such that the Birch and Swinnerton-Dyer conjectural formula predicts an odd prime divisor $\ell$ of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f)$ , but $\ell$ does not divide the modular degree of $A_f$ . These were taken from [AS05]. If there is an entry in the fourth column, this means we have verified the hypothesis of Theorem 5.4.2, hence there really is a nonzero element in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_f)$ that is not visible in $J_0(N)$ , but is strongly visible in $J_0(pN)$ . The notation in the fourth column is $(p,E,q)$ , where $p$ is the prime used in Theorem 5.4.2, $E$ is an elliptic curve, denoted using a Cremona label, and $q\neq p$ is a prime such that

$$\bigcap_{q'\leq q} {\mathrm{Ker}}(T_q'\vert _{\overline{W}} - a_{q'}(E)) = 0.$$

$A_f$	dim	$\ell\mid{\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f)_?$	moddeg	$(p,E,q)$ 's
551H	18	3	$2^{?} \cdot 13^2$	(2, 1102A1, -)
767E	23	3	$2^{34}$	(2, 1534B1, 3)
959D	24	3	$2^{32}\cdot 583673$	(2, 1918C1, 5), (7, 5369A1,2)
1337E	33	3	$2^{59} \cdot 71$	(2, 2674A1, 5)
1339G	30	3	$2^{48} \cdot 5776049$	(2, 2678B1, 3), (11, 14729A1,2)