Properties of the elliptic curve $y^2 \equiv x^3 – 2 \pmod 7$

Question

Can someone help me:

1) to list the points on the elliptic curve $E: y^2\equiv x^3 – 2\pmod 7$.

2) to find the sum $(3, 2) + (5, 5) $ on $E$.

Answer 1

For the first part, you compute the quadratic residues $\pmod 7$ : $$0,1,4,3^2 \equiv 2,4^2 \equiv 2,5^2 \equiv 4, 6^2 \equiv 1.\tag{A}$$

By simple calculations (by hand), you know that $x^3-2$ is not a quadratic residue for $x=0,1,2,4$. Then you have 7 points on $E$ :

$$(3,2) \quad;\quad (3,5) \quad;\quad (5,2) \quad;\quad (5,5) \quad;\quad (6,2) \quad;\quad (6,5) \quad;\quad \infty\tag{B}$$

---

For the second part, first compute the slope $$m = \dfrac{5 - 2}{5-3} = 3 \cdot 2^{-1} \equiv 3\cdot4 \equiv \color{purple}5 \pmod 7 \tag{C}$$ Then the sum of the 2 points (that lie on $E$) is

$$(\color{orange}3,\color{blue}2)+(\color{green}5,5) = (x = \color{purple}5^2 - \color{orange}3 - \color{green}5, y = \color{purple}5 \cdot (\color{orange}3-x) - \color{blue}2) \equiv (3,5\cdot0 -2) \equiv (3,5)\tag{D}$$