# Blog #1: The Sections of y^2=x^3-t^n x-t^n

This research is a continuation on Professor Lisa Fastenberg’s research, “Mordell-Weil groups in procyclic extensions of a function field.” In her research, she discovered that the Mordell-Weil group of the sections of  have a maximum rank of 56 when n is a multiple of 2520. These sections are pullbacks of sections for   when n = 1, 2, 3, 7, 8, 10, 12, 15, 18, 20, 42. However, we currently lack an explicit list of such sections, and computations using conventional methods are difficult for larger values of n. Therefore, the goal of this research is to write a computer program on Maple to find all sections of , and show that these sections are generators of the Mordell Weil Group.

Currently, I am learning the Maple programming language and doing preliminary testing on these elliptic curves:

# R

k, k’ Minimal Model at 0 Sections: (???)
1 1  y^2=x^3-xt-t (-1,i)
2 1  y^2=x^3-xt^2-t^2  (t, it) or (0, it)
3 2  y^2=x^3-xt^3-t^3  (t, it^2)

As I become more experienced in Maple, I will hopefully be able to replicate Professor Fastenberg’s previously computed sections, and continue to make further progress past what she did.