Elliptic Curve 37a1 (reviewed)

This elliptic curve $E$ has the least conductor of any elliptic curve over $\Q$ that is the sole member of its isogeny class.

This elliptic curve is associated to the [Somos-4 sequence] $\{a(n)\}$. Let $P$ be the generator $(0,0)$ of $E(\Q)$. Then for odd $n$ the $x$- and $y$-coordinates of $nP$ have denominators $d_n^2$ and $d_n^3$ where $$d_n = 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209$$ for $n=1,3,5,\ldots,19$, and $d_{2n-3} = a(n)$ in general, satisfying the Somos-4 recurrence $$ d_n d_{n+4} = d_{n+1} d_{n+3} + d_{n+2}^2. $$ The regulator of $E$, which is equal to the canonical height $\hat h(P) \approx 0.0511$, controls the growth of the $a(n)$: asymptotically $\log a_n \sim 2 \hat h(P) n^2$.

The integral points on $E: y^2+y=x^3-x$ correspond to solutions of the classical problem of finding all integers that are simultaneously the product of two consecutive integers and the product of three consecutive integers [since $y^2+y=y(y+1)$ and $x^3-x = (x-1)x(x+1)$]. That $210 = 5 \cdot 6 \cdot 7 = 14 \cdot 15$ is the last such example follows from the fact that $(0,0)$ generates the group of rational solutions; see page 275, exercise 9.13 of The Arithmetic of Elliptic Curves [10.1007/978-0-387-09494-6].