Further update: **gp** now has a command **ellrank** that finds
the rank quickly:

    allocatemem(2^26)
    E = ellinit([0,1,0,-15662264585,746984342506759]);
    ellrank(E)

takes under a second on my laptop to reply

    [8, 8, 0, [[78077, 294276], [116479, 22427332], [-1085129/9, 802159084/27], [-3855/49, 9382266704/343], [11662649/121, 15339215952/1331], [42354193/289, 196480125128/4913], [2521274/25, 1737395457/125], [112761813/1444, 19048354703/54872]]]

The first two numbers $r_1,r_2$ in the output are lower and upper bounds
on the rank; here $r_1 = r_2 = 8$ so this curve has rank 8.
The third number encodes information about the Tate-Shafarevic group Sha; 
here it says that Sha[2] is trivial.
The last output is an LLL-reduced $\bf Z$-basis $B$
for a rank-$r_1$ subgroup of the Mordell-Weil group E(Q).

This function was added by Bill Allombert in early 2021 according to
<https://pari.math.u-bordeaux.fr/archives/pari-dev-2102/msg00003.html>,
but I only learned of it last month from John Cremona.
The online documentation at
https://pari.math.u-bordeaux.fr/dochtml/html/Elliptic_curves.html
says that the 2-Selmer rank is computed unconditionally,
which would mean that the bounds $r_1,r_2$ do *not* depend on GRH or on
any other unproved hypothesis.  

$\langle B \rangle$ is *not* necessarily saturated, 
and indeed in this case the function **ellsaturation** finds a $\bf Z$-basis 
$B'$, again LLL-reduced, for a larger group of rational points 
that contains $\langle B \rangle$ with index 5:


    [[116479, 22427332], [17023, -22029500], [-139583, 14616304], [52797, 8199964],
    [-143366, 6761923], [31635, -16827628], [66259, 341548], [66305, 49728]]

Unlike **mwrank**, this program will not certify that
the result is fully saturated:
**ellsaturation** requires an upper bound on the saturation primes.
Still it takes about a second to certify that the index is not divisible by
any prime less than $2^{12}$; so it's morally certain ---
and likely provable --- that $\langle B' \rangle$
is the full Mordell-Weil group.