Finding the rank of certain Mordell curves

Question

I am interested in the elliptic curve $y^2 = x^3 + 16r^2$ for various cubefree $r$. (This is related to the elliptic curve $y^2 + ry = x^3$, sums of two rational cubes A159843, and the recent results of Alpöge, Bhargava, & Shnidman.)

For example, let $r=54919$. The analytic rank is 2 but I can't prove that its rank is > 0.

Are there any tools I can use to compute these, any places I can look them up, or any good theory to attack them? I know very little about elliptic curves, unfortunately. The curves of interest to me all have large conductors (~$10^9$).

• LMFDB has a database of elliptic curves but I can't seem to find the ones I'm looking for there, and can't even search it very well (only by computing some invariants* and searching through the resulting lists).
• Cremona's Elliptic Curve Data is a wealth of information, but it only has data for conductors less than half a million, so there's nothing there for me.
• PARI/GP has pretty good ability to work with elliptic curves and supplies ellrank as well as ellanalyticrank, but these run out of gas eventually.
• mwrank is now only available as eclib, and I haven't been able to build it (it ask me to recompile with -fPIC, and I'm trying, but it's not working).

* There are some easy values: any curve in this family has $j$-invariant 0 and discriminant $-432r^4$.

Answer 1

Magma is fairly good at computing Mordell–Weil groups of elliptic curves and sometimes succeeds where other libraries fail. For example, it successfully computes the Mordell–Weil group of the curve you mentioned in about 13 seconds on my laptop (running Magma 2.27-1):

E := EllipticCurve([0, 16*54919^2]);
A, mw, rank_proven, saturation_proven := MordellWeilGroup(E);
[mw(A.1), mw(A.2), mw(A.3)];

The booleans rank_proven and saturation_proven are both true, which means the full Mordell–Weil group has been provably computed. The first generator is the 3-torsion point $(0 : 219676 : 1)$, generating the torsion subgroup. The other two generators, generating the $\mathbb{Z}^2$ part, are $$(-74715322416/21390625 : -7431687117209948/98931640625 : 1)$$ and $$(\tfrac{-223505436219321136916938970992760345077374165859681250008575245328}{145217979187124628448668425810330104026209934618656495346234121} : \tfrac{-369619409961856952324804801809068062165946757491071404180443310935789117344859520908282493098069772}{1749969918763739600846483096788677644836465031382404342922206701087180078045471050628765268581} : 1).$$ Of course, as you said, such methods can only go so far before the computation becomes infeasible, but perhaps this will work for more of the particular curves of interest to you.