= Solution
For
$$
E:y^2=x^3+x^2+7x,
$$
use <two-isogeny descent> through
$$
E':y^2=x^3-2x^2-27x.
$$
The square-class maps send a nonexceptional point to the class of its $x$-coordinate. On $E$, possible classes are $\pm1,\pm7$; the defining quartics and positivity exclude the negative classes, while $(1,3)$ and $(7,21)$ realize $1$ and $7$. Thus the image has order two. On $E'$, the possible classes are $\pm1,\pm3$; the classes $1$ and $-3$ occur, while the quartics for $-1$ and $3$ have no primitive solution modulo $16$. Hence this image also has order two. The two-isogeny descent formula
$$
2^{r+2}=|\operatorname{im}\alpha|\,|\operatorname{im}\alpha'|
$$
therefore gives $r=0$.
The displayed curve has good reduction at $5$ and $11$, where direct counting gives
$$
\#E(\mathbb F_5)=6,
\qquad
\#E(\mathbb F_{11})=18.
$$
Reduction injects rational torsion of order prime to these characteristics, so its order divides $\gcd(6,18)=6$. We already have the six distinct points
$$
O,quad(0,0),quad(1,\pm3),quad(7,\pm21).
$$
Since the rank is zero, these are all the rational points and $E(\mathbb Q)\cong\mathbb Z/6\mathbb Z$.
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