About the problem
For each squarefree polynomial \(f(x) \in \mathbb{Z}[x]\) of degree \(6\), the curve \(C_f\) defined by \(y^2 = f(x)\) is hyperelliptic of genus 2 and thus has finitely many rational points by Faltings’ theorem. That is, there are finitely many \(x \in \mathbb{Q}\) so that \(f(x)\) is a square.
This problem asks for a genus 2 curve with more than 642 rational points. There is a genus 2 curve with 642 rational points: the curve
\[y^2 = 82342800 x^6 - 470135160 x^5 + 52485681 x^4 + 2396040466 x^3 + 567207969 x^2 - 985905640 x + 247747600,\]due to Elkies and Stoll in 2008, and this is the current record. If a solution to this problem is found, then it would beat this record from 2008.
It was conjectured by Caporaso, Harris, and Mazur that for every genus \(g\) there is a uniform upper bound \(f_K(g)\) such that every genus \(g\) curve over the number field \(K\) has at most \(f_K(g)\) rational points. This conjecture is open even in the case of genus \(2\) curves over the rationals, i.e. even in the setup of this problem. A solution to this problem would ideally help us understand this conjecture, although there is also a small risk that if the conjecture is true, then the problem is unsolvable.
Prompt
For each squarefree polynomial f(x) in Z[x] of degree 6, the curve
C_f : y^2 = f(x)
is hyperelliptic of genus 2. By Faltings' theorem (the Mordell conjecture), a curve of
genus at least 2 over Q has only finitely many rational points. Equivalently, there are
only finitely many x in Q for which f(x) is the square of a rational number.
The rational points of C_f come in two kinds:
- Affine points: each x in Q with f(x) = y^2 != 0 yields a pair (x, y) and (x, -y); an
x with f(x) = 0 yields the single point (x, 0).
- Points at infinity: the smooth projective model has two points at infinity. They are
rational if and only if the leading coefficient c6 of f is the square of a rational
(equivalently, a perfect-square integer).
The current record is 642 rational points, achieved in 2008 by Elkies and Stoll with
y^2 = 82342800 x^6 - 470135160 x^5 + 52485681 x^4 + 2396040466 x^3
+ 567207969 x^2 - 985905640 x + 247747600.
(Its leading coefficient 82342800 is not a perfect square, so this curve has no rational
points at infinity and all 642 of its rational points are affine.)
## Task
Find a squarefree polynomial f(x) in Z[x] of degree exactly 6 such that the curve
y^2 = f(x) has MORE THAN 642 rational points in total (affine points together with any
rational points at infinity).
(There is no loss of generality in taking integer coefficients: f and c^2 * f define
isomorphic curves for any c in Q*, so denominators and square factors of the leading
coefficient can be cleared.)
## Submission
Write your answer to a file, then submit the path to that file using the `submit_answer`
tool. The file must contain a single JSON object with exactly two keys, "coeffs" and
"points":
- "coeffs": the 7 coefficients of f as a list of integer strings, CONSTANT TERM FIRST,
i.e. ["c0", "c1", "c2", "c3", "c4", "c5", "c6"] meaning f(x) = c0 + c1 x + ... + c6 x^6.
Each coefficient must be an integer written as a string (e.g. "0", "-7", "12"). The
leading coefficient c6 must be nonzero, so that f has degree exactly 6.
- "points": a list of [x, y] pairs giving the AFFINE rational points, each coordinate a
string parseable as an exact rational (an integer like "5" or a fraction like "-3/2").
Every pair must satisfy y^2 = f(x). Do NOT list the points at infinity -- the verifier
adds them automatically (two of them when c6 is a perfect square, otherwise none).
The file should contain exactly this JSON object and nothing else. The example below only
illustrates the format; its points do not give a record curve:
{"coeffs": ["1", "0", "1", "0", "1", "0", "1"], "points": [["0", "1"], ["0", "-1"], ["1", "2"], ["1", "-2"]]}
Everything is checked exactly with arbitrary-precision rational arithmetic, so approximate
or floating-point values will not be accepted.