I am trying to do some elliptic curve calculations by hand, just to refresh myself on how the system works. I calculated some points and did some operations by hand. I am trying to double check my work with some web calculators, but I have tried several different calculators and each gives me a different result...
Is there a verified implementation or library somewhere that I could use, rather than pet projects that people have on their websites?
I am working with the curve
$y^2 \equiv x^3 - 2 \pmod 7$
I calculated the points to be
I calculated $(3,2) + (5,5) = (3,5)$ I'm fairly confident in this answer.
The problem I'm struggling with however is $(3,2) + (3,2) = (3,5)$. I have tried several times and have gotten the slope between the two points to be $27/4$. When I convert this to an integer $\mod 7$, I get the slope equal to either $3$ or $5$. When $m=3$, I get the point $(3,5)$. When $m=5$, I get the point $(5,1)$, which isn't on the curve, so I'm assuming I'm doing this step incorrectly.