I am guessing $\mathfrak p=(x^5+x^3y^2+x^2y^3+y^5-y)$ is a prime ideal. Here is my attempt:
Every $y^n$ term for $n\geq 5$ can be reduce to a lower degree modulo $\mathfrak p$ by using $$y^5 = y-x^2y^3-x^3y^2-x^5$$ so that under modulo $\mathfrak p$ every polynomial is of the form $$y^4p_4(x)+y^3p_3(x)+y^2p_2(x)+yp_1(x)+p_0(x)$$ but this is all I have got.
I am guessing $$\Bbb Q[x,y]/\mathfrak p\simeq \Bbb Q[x]\oplus \Bbb Q[x]\oplus \Bbb Q[x]\oplus \Bbb Q[x]\oplus \Bbb Q[x]$$