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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 are already frames in terms of finite curves.)

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.

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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?

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  • $\begingroup$ I don't have a formal notion of "natural" in mind---put another way, I mean something like "not something that appears transparently reverse-engineered for this purpose." But certainly "when is the product of two consecutive integers equal to the product of three consecutive integers" is quite natural by my reckoning. (+1) $\endgroup$ – Travis Willse Sep 21 at 5:25
  • $\begingroup$ This got me curious: Via the evident symmetry $(x, y) \leftrightarrow (x, -y - 1)$ it suffices to restrict to $y \geq 0$. Excluding the trivial solutions where one $x - 1, x, x + 1$ is $0$ or $1$, the SAGE function IntegralPoints() reports that the only other solution is $5 \cdot 6 \cdot 7 = 14 \cdot 15$. (Probably this result can be derived by elementary means, too.) $\endgroup$ – Travis Willse Sep 21 at 5:40
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I want to describe a different kind of a context – algebraic geometry codes. I'm not sure I would call this natural though. Also, the interest is then only on a (good) reduction of the curves modulo a single prime $p$. Implying that we lose nearly all of the information about the identity of the curve. After all, many drastically different elliptic curves share the same reduction modulo $p=2$ :-/

The game in this application is to look for curves defined over a finite field $\Bbb{F}_q$ such that they have as many rational points as possible for a code of a prescribed genus. A larger number of points allows us to build longer codes without paying the price of a genus penalty on the error-correcting capability. The widely used Reed-Solomon codes (appear for example in CD-ROMs and Quick Response -codes are based on $g=0$ curves. In a sense $g=1$ curves are the next best thing, and may be what an application needs, if we need more than $q+1$ points rational over the field $\Bbb{F}_q$.

Anyway, all the four curves you listed have a good reduction modulo two, and become isomorphic to $$ E:y^2+y=x^3+x $$ modulo $p=2$. Simple counting shows that $\#E(\Bbb{F}_2)=5$, implying that the zeros of its $\zeta$-function are $\alpha=\alpha_{1,2}=-1\pm i$. Here $\alpha^4=-4$ is real and negative implying that the Hasse-Weil bound $$ \#E(\Bbb{F}_q)\le q+1+2\sqrt q $$ is met with equality for these curves whenever $q=2^n$, $n\equiv4\pmod8$. Meaning that for those fields this curve may be an attractive choice.

But, this is not very exceptional. Curves with $\pmod 2$ reduction $y^2+y=x^3$ are at upper limit of the Hasse-Weil bound, when $q=2^n, n\equiv2\pmod4$, and the curves that reduce to $y^2+y=x^3+x+1$ also achieve Hasse-Weil with equality whenever $n\equiv4\pmod8$. This is not surprising for the left hand side, $y^2+y$, means that modulo $p=2$ we are looking at an Artin-Schreier extension, when the number of points comes from an additive character sum. In characteristic two the trace of a cubic is really a quadratic form in disguise, and the appearance of a quadratic form severely restricts the range of values of the relevant character sums. The theoretical excitement about AG-codes came largely from sequences of curves $C_i$ with increasing values of $g_i$ such that we get asymptotically good ratios $g_i/\#C_i$.

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  • $\begingroup$ This is quite interesting, but like you say this particular construction produces the same curve over $\Bbb F_{2^n}$ for a /lot/ of rational elliptic curves---(including) any with integral coefficients with parities the same as in the examples. Are there very large primes $p$ for which any of these curves have a good reduction and have point counts the floor of the upper Hasse-Weil bound? That still wouldn't give uniqueness, of course, but the the relative rarity of such curves would still make these curves interesting examplars. $\endgroup$ – Travis Willse Sep 21 at 17:36
  • $\begingroup$ (In any case, +1, including for exposing all of the ideas here---I'm a differential geometer, not a number theorist, but the topic of elliptic curves is a personal favorite.) $\endgroup$ – Travis Willse Sep 21 at 17:37

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