Can anyone help me to compute how many such homomorphisms (with the domain and codomain shown below) exist? 
2026-03-26 22:15:06.1774563306
How many homomorphisms from this quotient ring to Z7?
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Any $\mathbb{Z}$-module homomorphism from $\frac{\mathbb{Z}[x,y]}{x^3+y^2-1}$ to $\mathbb{Z}/(7)$ depend on the choices for the image of $x$ and $y$ in $\mathbb{Z}/(7)$. Moreover the image of $x$ and $y$ have to satisfy the equation $x^3-y^2+1=0$ in $\mathbb{Z}/(7)$. These are the ordered pairs $(0,1)$, $(0,6)$, $(1,0)$, $(2,0)$, $(4,0)$, $(5,3)$, $(5,4)$, $(6,3)$, $(6,4)$.