Particular (rational) elliptic curves arise in many contexts outside the study of elliptic curves themselves. For example, this solution to this question asking which squares of triangular numbers $T(k)$ are themselves triangular numbers proceeds by applying a suitable change of coordinates $(k, n) \rightsquigarrow (U, V)$ the elliptic equation $T(n) = T(k)^2$ into the form $V^2 = q(U)$ for a quartic polynomial $q$, after which we can use an algorithm of Tzanakis (and integral version of the so-called LLL reduction) to find all of the integer solutions $(U, V)$, and hence (because of the form of the coordinate transformation) all of the integer solutions $(k, n)$. The elliptic curve defined by the equation here is curve $\texttt{192a2}$ in Cremona's tables of elliptic curves with small conductor.
Distinguished among rational elliptic curves are the three (isogenous) curves of the smallest realized conductor, $11$. These are, are up isomorphism (the given concrete curves are the minimal models):
\begin{array}{cl}
\texttt{11a1} & y^2 + y = x^3 – x^2 – 10 x – 20 \\
\texttt{11a2} & y^2 + y = x^3 – x^2 – 7820 x – 263580 \\
\texttt{11a3} & y^2 + y = x^3 – x^2
\end{array}
In what contexts outside the direct study of elliptic curves do (any of) these curves occur (up to isomorphism) naturally, analogously to the way that $\texttt{192a2}$ occurs in the above problem concerning polygonal numbers?
(A handful of answers elsewhere on the site reference these curves, but only in questions that concern curves over finite fields.)
Already the conductor (192) in the example above is relatively small—fewer than 700 curves have a smaller conductor. One can inspect the elliptic curves that arise in the analogous problems of which squares of $m$-gonal numbers are squares of other $m$-gonal numbers, but for $3 \leq m \leq 16$ (excluding $m = 4$, which gives rise to a genus-zero equation with obvious solutions), $192$ is the smallest occurring conductor. (In fact, the curve $\texttt{192a2}$ appears twice in this context, up to isomorphism: In the above case, $m = 3$, and in the case $m = 6$ of hexagonal numbers.)
It's plausible (at least to a non-[number theorist] like me) that the fact all that three of the conductor-$11$ elliptic curves have rank zero might thwart their occurrence in interesting places elsewhere. If that's the case (or even if not), that suggests a natural next question:
In what contexts does the elliptic curve $\texttt{37a}$ ($y^2 + y = x^3 – x$)—the unique rational elliptic curve of rank $1$ of minimal conductor—occur naturally?
Reference
Tzanakis, N. "Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations." Acta Arithmetica 75 (1996), 165-190.
Best Answer
Regarding 37a: When is the product of two consecutive integers, $y$ and $y+1$, equal to the product of three consecutive integers, $x-1$, $x$, and $x+1$.
Is that natural? It's the sort of question one might generalize from $y^2 = x^3$, which is addressed on this site, in which we repeat numbers rather than iterate. What's your notion of naturality?