$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves

Question

Do there exists rational numbers $x$ and $y$ such that $$ y^3 = x^4 + x + 2 ? $$

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 rank can be computed using RankBounds function in Magma, and this particular curve has rank 0. There is a magma function (Chabauty0) for computing rational points on rank 0 hyperelliptic curves, but Picard curves are not hyperelliptic. On the other hand, Hashimoto and Morrison https://arxiv.org/abs/2002.03291 recently showed how to compute rational points on Picard curves of rank 1. Based on this, it looks like the case of Picard curves of rank 0 should be tractable, but I cannot find any reference, and also cannot solve this myself, hence the question. The particular equation above is chosen because it is the simplest example of such curve for which there is a rational point everywhere locally but there are no obvious rational points.

Answer 1

The below Magma code determines the size of $J_C(\mathbb{F}_p)$ for various primes $p$, and finally compute the GCD of their orders, which gives you a bound on the size of $J_C(\mathbb{Q})$. For a discussion of this, see Section 4.1 of this paper. To determine $J_C(\mathbb{F}_p)$, you can just evaluate the numerator of the zeta function of the reduction of $C$ mod $p$ (see e.g., Hindry--Silverman Diophantine Geometry Exercise C.4).

The below code does this for primes up to 30 and tells you that the GCD of $\#J_C(\mathbb{F}_p)$ for these primes is 1, and hence $J_C(\mathbb{Q})$ is just the trivial group. The code is general and can work for any equation of this form of rank 0. If you do have some non-trivial torsion in $J_C(\mathbb{Q})$, then you will need to work harder to determine the rational points.

I will also note that you can run this on the online magma calculator.

P2<x,y,z> := ProjectiveSpace(Rationals(),2);
C := Curve(P2,z*y^3 - (x^4 + x*z^3 + 2*z^4));
bad := {2,3};
bool := false; p := 2;
torsOrders := {@@};
while  p le 30 do
  p :=  NextPrime(p);        
  p := p in bad select NextPrime(p) else p;
  Cp := Curve(Reduction(C,p));
torsOrders := torsOrders join  {@ Evaluate(Numerator(ZetaFunction(Cp)),1) @};
end while;
GCD(torsOrders); //1