TITLE: Group structure associated to the equation $y^2 = x^2 + x + 1$
QUESTION [5 upvotes]: The elliptic curve group law allows us to add two solutions $(x_1, y_1), (x_2, y_2)$ of the equation $y^2 = x^3 + ax + b$ to obtain a new solution $(x_3, y_3)$. Is there an analogous group law associated to the equation $y^2 = x^2 + x + 1$?

REPLY [2 votes]: If $char(k)\ne 2$ then

$y^2=x^2+x+1$ is isomorphic through $x=z-1/2$ to $y^2=z^2+3/4$

Which is isomorphic through $y = \frac{u+v}{2},z=\frac{u-v}{2}$ to $uv=3/4$.

If $char(k)\ne 3$ then it is isomorphic to $uw=1$ through $v=\frac43w$

Which is the multiplicative group $GL_1(k)$ whose group law is $(u_1,w_1).(u_2,w_2)=(u_1u_2,w_1w_2)$


If $char(k)=2$ then through $y=t-x$ the curve is isomorphic to $\Bbb{A}^1_k$ which is a group. If $char(k)=3$ then the curve is isomorphic to $y^2=z^2$ which is not irreducible, removing $(0,0)$ (the only singular point) it becomes $GL_1(k)\cup GL_1(k)= GL_1(k) \times \pm 1$ which does have a group law.