Rational points of $v^2+v=u^3+u^2+u$ (the order-three diamond quotient of $X_1(19)$)

Every rational affine point $(u,v)$ on the elliptic curve

has $u=0$; equivalently its only rational points are $\infty$, $(0,0)$ and $(0,-1)$, so the Mordell–Weil group is $\mathbb Z/3\mathbb Z$ (rank zero).

This curve is the quotient of $X_1(19)$ by the order-three subgroup of diamond operators. In Xiang Huang's Lean development it is the only arithmetic input to the exclusion of rational points of order $19$: an explicit algebraic map sends a nondegenerate solution of the order-19 Tate division equation to an affine point with $u\neq 0$.