During this period of time, I explored obtaining the coefficients of the group, y^2=x^3-t^n x-t^n. I specifically explored the groups when n = 3 and 1. I attempted to obtain the coefficients of the group y^2=x^3-t^3 x-t algebraically, by replacing at+b for x and ct^2+dt+e for y.
This resulted in (ct^2+dt+e)^2 = (at+b)^3-(at+b)t^3-t, obtaining a^3t^3 + 3a^2bt^2 + 3ab^2t + b^3 – at^4 – bt^3 – t
By doing this, we know that c^2 = -a, e^2 = b^3, 2cd = a^3-b, 2ce+d^2=3a^2b, de=3ab^2-1
I attempted to solve for these values in maple by using the collect(simplify(equation, variable to solve for)) function to go further, since the variables cannot be simplified any further using basic algebraic methods. Maple has many useful functions that include the simplify, collect, and solve functions that allow one variable to be isolated. Then, we can set all instances of a variable, such as “c,” into what it is equal to. We know that a = -c^2, so all instances of a can be replaced with (-c^2).
I hope to continue doing this over Winter Break and apply this methodology to more complex functions. I also hope that I will be able to develop a more efficient method, possibly not using Maple, so that I can compute the larger coefficients.