Checking whether a point is on an elliptic curve

Question

I have the point $(2,1)$ and I would like to check whether it is on my elliptic curve.

The elliptic curve is defined by $y^2 = x^3 + 3x + 1$ in $GF(7)$.

My solution to check whether or not the point is on the curve was to substitute the point into the equation:

$$ (1)^2 = (2)^3 + 3(1) + 1. $$

From inspection one may observe the left side does not equal the right side, therefore, the point is not on the curve.

However, I suspect this is incorrect I was just looking for some guidance. Thank you so much.

Answer 1

Your solution is not correct. $$(1)^2 = (2)^3 + 3(2) + 1=1 (mod 7).$$

Answer 2

You have a small error there. Given the curve equation in $\operatorname{GF}(7)$ check the point $P(2,1)$

$$y^2 = x^3 + 3x + 1$$ put $x= 2, y=1$

\begin{align} (1)^2 &= (2)^3 + 3(\mathbf{\color{blue}{2}}) + 1 & & \pmod{7}\\ 1 &= 8 + 6 + 1 & &\pmod{7}\\ 1 &= 15 = 2\cdot7+1 & &\pmod{7}\\ 1 &= 1 & &\pmod{7}\\ \end{align}

Therefore, the point is on the curve.

Your mistake was $(1)^2 = (2)^3 + 3(\mathbf{\color{red}{1}}) + 1 \pmod{7}$