Pries: 405 Number Theory, Spring 2012. Homework 7.
Due: Friday 4/13.
Elliptic curves
Read: Stein chapter 6.
Use SAGE to complete 5 of the following problems.
1. 6.2, 6.3, 6.4, 6.10.
2. Factor 618240007109027021 by the Pollard p − 1 method.
3. Let E be the elliptic curve y 2 = x3 + 17. It has the following 5 rational points:
P1 = (−2, 3), P2 = (−1, 4), P3 = (2, 5), P4 = (4, 9), P5 = (8, 23).
(a) Show that each of P2 , P4 , and P5 is equal to mP1 + nP3 for some choice of integers
m and n.
(b) Compute the points P6 = −P1 + 2P3 and P7 = 3P1 − P3 .
(c) There is exactly one point P8 = (x8 , y8 ) on E (other than P1 , . . . , P7 ) such that
x8 and y8 are integers and y8 > 0. Find it.
(d) What is the rank of E?
1