Does this equation have a rational point? (Elliptic curve?)

Question

Can someone check pls if,

$$852 + 3017 x - 1104 x^2 + 2009 x^3 - 3362 x^4=y^2$$

has a rational point? (This arose in an equal sums of like powers problem.)

P.S. I've checked $x=p/q$ for $\text{Abs}\,(p,q)<200$, but didn't find anything. :(

Answer 1

There may be faster ways, but this is what I have found.

Assume $x=p/q$ with $p,q$ coprime integers and $q$ positive. Multiply the curve equation by $q^4$ to make its left-hand side an integer: $$F(p,q)=-3362 p^4 + 2009 p^3 q -1104 p^2 q^2 + 3017 p q^3 + 852 q^4=(q^2 y)^2$$ Therefore $q^2 y$ must be an integer to solve the equation, and the right-hand side must be a square in $\mathbb{Z}$.

Assume $5\mid q$, then $F(p,q)\equiv-2 p^4\pmod{5}$, which is not a square unless $5\mid p$, but the latter would contradict the lowest-terms requirement. So $5\not\mid q$, hence any solution's $x$ can be represented in $\mathbb{Z}_5$ (the $5$-adic integers), and $F(x,1)$ must be a square in $\mathbb{Z}_5$.

Using Pari/GP, I verified that $F(x,1)$ is never a square modulo $5^4$, so the original equation has no rational solutions.