= Solution
The <Weil pairing> $e_3:C[3]\times C[3]\to\mu_3$ is nondegenerate and Galois equivariant. If all of $C[3]$ is rational over a field, pairing a basis produces a primitive cube root of unity in that field. Thus a quadratic field with full 3-torsion must contain $\mathbb Q(\zeta_3)$ and must equal it.
More generally, if $E(\mathbb F_p)[\ell]\cong(\mathbb Z/\ell\mathbb Z)^2$, Frobenius acts as the identity on $E[\ell]$. Its characteristic polynomial $X^2-aX+p$ is therefore congruent to $(X-1)^2$ modulo $\ell$. Comparing coefficients gives
$$
a\equiv2\pmod\ell,
\qquad p\equiv1\pmod\ell.
$$
In particular $\#E(\mathbb F_p)=1+p-a$ with $a\equiv2\pmod\ell$.
Solved by gpt-5.6-sol high.
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