# Elliptic curves on finite fields

On page 22 it shows an eliptic curve over F17.

I have added the orange lines and labels.

Do the orange lines illustrate that a line through points at P(2,10) and Q(5,9) also intersects the "curve" at -R(12,1)?

i.e. Does the finite field version graphically match an elliptic curve over real numbers, with the provisos that the finite field curve is discontinuous and that lines wrap?

Or am I hopelessly confused about elliptic curves on finite fields (as I expect).

P.S. If this is an accurate graphical representation, how do I illustrate tangents to the discontinuous curve?

P.P.S. There are multiple lines which contain three points. X, Y and -Z, in purple, are another (below).

• Your graphs represent correctly lines over $\mathbb{F}_{17}$: They look as usual lines, but they "overflow" as you have shown. Unfortunately, I don't have a good idea about how to "plot" tangent lines in this case. It's very similar to the concept of derivative: you can extend its algebraic properties to the finite-field-case, but its visual interpretation is lost. – Daniel Jan 18 '19 at 14:27

You are correct about the graphical representation. A line in the finite field $$\mathbb{F}_p$$ is not $$y=ax+b$$ but "informally" $$y \equiv ax+b \pmod p$$, so the line "repeats" itself. For the tangent, you refer to the algebraic formula to trace the line. Have a look at the third figure here.