What are (a) integer, (b) rational solutions to the equation $$ y^3 = x^4 + x + 1 ? $$ There are obvious solutions $(x,y)=(-1,1)$ and $(0,1)$, are they the only ones?
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and these problems has been solved in a number of special cases. The "next simplest" case are Picard curves, which can be described by equations in the form $y^3=P(x)$, where $P(x)$ is a polynomial of degree 4. An important parameter measuring the complexity of this problem is the rank of the Jacobian of the curve. The bounds for rank can be computed using RankBounds function in Magma, and for this particular curve the lower and upper bounds are 1 and 2. The case of rank $r=0$ is solved in the answer to my previous question $y^3 = x^4 + x + 2$, and rational points on rank 0 Picard curves , the case $r=1$ is studied in https://arxiv.org/abs/2002.03291 , so the case $r=2$ is the next one to consider. The Chabauty--Kim method should theoretically work whenever $r<g$, and for this curve the genus $g=3$, so there is a hope. This particular equation is the nicest-looking representative from this family, hence the question.
Jacobian(C);
), Magma fails to obtain a genus one model for the curve and exits with a runtime error. $\endgroup$