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

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.

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.