How do I show that two points are on an elliptic curve?

Question

How do I show that two points are on an elliptic curve - or not on the curve?

$$E\left(\mathbb{F}_{23}\right)=\left\{(x, y) \in \mathbb{F}_{23} \times \mathbb{F}_{23}: y^{2}=x^{3}+x-11\right\}\;\quad P := (0,4)\quad Q := (2,5).$$

Answer 1

As Angina Seng wrote here: $$y^2 = x^3 + ax + b$$ Now place the coordinates P(x,y) into the formula. Don't forget that both sites are modulo p (our Field order).

$$0^2 \pmod{23} = (4^3 + 4 - 11) \pmod{23} \rightarrow 0 \neq 11$$ The first point P is not on the curve.

$$2^2 \pmod{23} = (5^3 + 5 - 11) \pmod{23} \rightarrow 4 = 4$$ The second point is on the curve.