Large initial solutions to $x^3+y^3 = Nz^3$? Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to,
$$x^3+y^3 = Nz^3\tag1$$
for $N=94$ is,
$$15642626656646177^3 + (-15616184186396177)^3 = 94\cdot 590736058375050^3\,?$$
If not, then what is the largest initial solution for $N<100$? Or $N<200$?
P.S. Related posts are $x^3+y^3 = 6z^3$, and $x^3+y^3 = 22z^3$, and $x^3+y^3 = 313^2z^3$. See also this paper by Dasgupta and Voight for more details (including the elliptic curve for eq.1).

After some insight courtesy of Achille Hui, I was able to answer my own question. It is well known (see also this) that $x^3+y^3=N$ is birationally equivalent to the elliptic curve $u^3-432N^2=v^2$ using the transformation $x=\frac{36N+v}{6u}$, $y=\frac{36N-v}{6u}$.
In this comment, Hui suggested the command,

Q$<x>$ := PolynomialRing(Rationals()); E00 := EllipticCurve(x^3-432*94^2); Q00 := Generators(E00); Q00;

on the Magma online calculator. We then find,

(62511752209/2480625 : -15629405421521177/3906984375 : 1) 

Substituting this onto the transformation, we get,
$$x = \frac{15642626656646177}{590736058375050}\\
y = \frac{-15616184186396177}{590736058375050}$$
thus Magma confirms the solution given in the original question is indeed the smallest.
P.S. Incidentally, the numerators nicely factor as,
$$15642626656646177-15616184186396177 =2^4\cdot 3^8\cdot5^6 \cdot 7^3\cdot47$$  
