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?



Mordell curves with large rank
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(This superseded a rank-16 pair from earlier that month, and curves of ranks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].

```

