Level 959

We do similar computations for a $24$ -dimensional abelian subvariety of $J_0(959)$ . We have $959=7\cdot 137$ , which is square free. There are five newform abelian subvarieties of the Jacobian, $A_2, A_7, A_{10}, A_{24}$ and $A_{26}$ , whose dimensions are the corresponding subscripts. Let $A_f = A_{24}$ be the 24-dimensional newform abelian subvariety.

Proposition 6.2.1   There is an element of order 3 in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A_f/\mathbb{Q})$ which is not visible in $J_0(959)$ but is strongly visible in $J_0(2 \cdot 959)$ .

Proof. Using Magma we find that $m_A = 2^{32} \cdot 583673$ , which is coprime to $3$ . Thus we apply Theorem 5.4.2 with $\ell=3$ and $p=2$ . Consulting [Cre] we find the curve E=1918C1, with Weierstrass equation

$$y^2 +xy + y = x^3 - 22x - 24,$$

with Mordell-Weil group $E(\mathbb{Q})\cong \mathbb{Z}\oplus \mathbb{Z}\oplus (\mathbb{Z}/2\mathbb{Z})$ , and

$$c_2 = 2, c_7 = 2, c_{137}=1, \overline{c}_2 = 6, \overline{c}_7 = 2, \overline{c}_{137}=1.$$

Using [Cre] we find that $E$ has no rational $3$ -isogeny. The modular form attached to $E$ is

$$g = q - q^2 - 2q^3 + q^4 - 2q^5 + \cdots,$$

and we have

$$\det(T_2\vert _{\overline{W}} - (-2)) = 2177734400 \equiv 2 \pmod{3},$$

which completes the verification. $\qedsymbol$