= Solution
The <elliptic-curve discriminant> is supported at $2$ and $3$, so $5$ and $7$ are primes of <good reduction>. Direct point counts give
$$
\#\widetilde E(\mathbb F_5)=8,
\qquad
\#\widetilde E(\mathbb F_7)=8.
$$
For example, summing $1+\chi_p(x(x+1)(x+4))$ over $x\in\mathbb F_p$ and adding the point at infinity gives these values.
The <reduction of torsion points on an elliptic curve> at the two primes shows that $|E(\mathbb Q)_{\mathrm{tors}}|$ divides eight. Part (a) shows that $P_2$ has order four, and $P_1$ is an independent point of order two because it does not lie in $\langle P_2\rangle$. They already generate eight points, so
$$
\boxed{E(\mathbb Q)_{\mathrm{tors}}=\langle P_2\rangle\oplus\langle P_1\rangle
\cong\mathbb Z/4\mathbb Z\oplus\mathbb Z/2\mathbb Z.}
$$
Back to article page
