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In this post, we considered the elliptic curve,

$$x\big(x+(m-1)^2\big) \big(x+(m+1)^2\big) = y^2\tag1$$

Online Magma was able to solve the case $m=2^7$ but timed-out on $m=5^7$,

$$x\big(x+78124^2\big) \big(x+78124^2\big) = y^2$$

However, Allan MacLeod in this answer was able to find a rational point on it.


This paper "Finding rational points on elliptic curves using $6$-descent and $12$-descent" (2007) by Fisher includes the elliptic curve,

$$x\big(x+(n-2)^2\big) \big(x+(n+2)^2\big) = y^2\tag2$$

This has at least five torsion points,

$$u = 0,\quad -(n-2)^2,\quad -(n+2)^2,\quad \pm(n-2)(n+2)$$

I tried the case $n=809$ of $(2)$,

$$x\big(x+807^2\big) \big(x+811^2\big) = y^2$$

on online Magma using the commands,

Q<x> := PolynomialRing(Rationals()); 
n:=809;
E00 := EllipticCurve(x *(x + (n + 2)^2)*(x + (n - 2)^2)); 
Q00 := Generators(E00); 
Q00;

and just got [ (-651249 : 0 : 1), (654477 : 1058943786 : 1) ] which are two of the torsion points.

However, according to the paper, $n=809$ does have a non-torsion rational point of huge height.

Q: What are the assumptions Magma uses to find the non-torsion point of $m=2^7=128$, but not $n=809\,$? Why didn't it just time-out like the case $m=5^7=79125$, or is it a bug?

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Generators(E00) will give you generators of the Mordell-Weil group of E00 that it computes. However, Magma cannot compute the Mordell-Weil group for this curve: if you ask for MordellWeilGroup(E00) it returns:

> MordellWeilGroup(E00);
Abelian Group isomorphic to Z/2 + Z/4
Defined on 2 generators
Relations:
    2*$.1 = 0
    4*$.2 = 0
Mapping from: Abelian Group isomorphic to Z/2 + Z/4
Defined on 2 generators
Relations:
    2*$.1 = 0
    4*$.2 = 0 to Set of points of E00 with coordinates in Rational Field given 
by a rule [no inverse]
false false

These final two return values of false mean, respectively, that the rank of the group it computes is not known to be the rank of the curve, and that the group it computes is not known to be the full Mordell-Weil group.

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  • $\begingroup$ Thanks. This gives me yet another reason to be careful when dealing with CAS. Using the command Generators(), it would have been better if the output is the actual generators (like for $m=128$), or a humble statement "cannot compute" (like for $n=809$). $\endgroup$ Jan 9, 2018 at 4:18
  • $\begingroup$ You should definitely check the documentation to see what methods are used, especially since there is no known general algorithm to compute the MordellWeilGroup, and finding partial results is useful. You can adjust optional parameters Effort and HeightBound to tell Magma to be more thorough (by default only 2-descent and 4-descent are used, and it only searches up to height of 15). $\endgroup$ Jan 9, 2018 at 7:03

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