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An elliptic curve is (for the purpose of this question) a cubic algebraic curve defined by an equation (short Weierstrass equation) of the form

$$\displaystyle E_{a,b} : y^2 = x^3 + ax + b, a, b \in \mathbb{Z}, 4a^3 + 27b^2 \ne 0.$$

The elliptic curves with $a = 0$ are called Mordell curves.

Do we have examples of Mordell curves with large algebraic rank, say exceeding 10?

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As far as I know, this record is still held by a 3-isogenous pair of Mordell curves of rank $\bf 17$ that I found and announced in February 2016. (This superseded a rank-16 pair from earlier that month, and curves of ranks 13, 14, 15 from October 2009.)

The curves $y^2 = x^3 + b$ and $y^2 = x^3 - 27 b$ are always related by an isogeny of degree $3$, and in particular have the same rank. The record $b,-27b$ have $$ b = -908800736629952526116772283648363 $$ (in factored form, $-2195745961 \cdot 413891567044514092637683$). The curve $y^2 = x^3 - 27 b$ has $17$ independent points

[-110315760690, 152299457785937151],
[-218829008658, 118569576333381183],
[194693247690, 178654854781822599],
[-12083686365, 156639252691623474],
[179588218407, 174154202398188288],
[660796972800, 559532270810391651],
[481938369495, 369425010854453724],
[532637728899, 419104420151289750],
[891937317975, 856808203106532276],
[1556910033324, 1948958451538253955],
[1369152212199, 1609695603071293320],
[-249954149276, 94452185380426435],
[527526224524, 413931980240076925],
[2095375244992, 3037184017947911267],
[3020920353232, 5252935870900542563],
[45908680009155, 311058636438867847974],
[209109621212430, 3023855428577131273599].
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    $\begingroup$ Where did you initially announce this result? $\endgroup$
    – Timothy Chow
    Commented Mar 12, 2023 at 1:26
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    $\begingroup$ Offline, Noam told me that the rank-16 pair was announced on the NMBRTHRY mailing list but he couldn't quite recall exactly how/where he announced the rank 17 example above. $\endgroup$
    – Timothy Chow
    Commented Mar 13, 2023 at 15:43
  • $\begingroup$ I received an announcement of the rank 17 curve in an email titled "j=0, rank 17" from Noam on Feb 23, 2016 (I was one of a dozen recipients). This document is dated Feb 18, 2016 math.harvard.edu/~elkies/many_pts_asu.pdf $\endgroup$
    – duje
    Commented Mar 17, 2023 at 18:36
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Yes. Apparently Tom Womack used to maintain a webpage devoted to Mordell curves of high rank at the (no longer functional) page here although it can be viewed with the help of the Wayback Machine. This reports that the curves with $$ k = 35470887868736225 \text{ and } k = -46111487743732324 $$ have rank $11$, while the curve with $$ k = -6533891544658786928 $$ has rank $12$. This rank 12 curve was found by Jordi Quer in connection with searching for imaginary quadratic fields of high $3$-rank (see the paper ''Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12'' in C. R. Acad. Sc. Paris. from 1987). Quer reports two more rank 12 Mordell curves in this paper.

Elkies and Rodgers also found a rank $11$ curve of the form $x^{3} + y^{3} = k$ (isomorphic to $y^{2} = x^{3} - 432k^{2}$) in their 2004 ANTS paper.

I do not know if any of these results have been improved recently.

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