Project: To compute the rank of the MordellWeil group of the Jacobian of the genus 2 curve
y^{2} = x^{6}  2x^{5}  x^{4} + 4x^{3} + 3x^{2} + 2x + 1
over Q. If there is extra time, one can try also to find all the rational points on the curve, compute the rational torsion subgroup of the Jacobian, or the endomorphism ring of the Jacobian.
References:

Flynn, Poonen, Schaefer, "Cycles of quadratic polynomials and rational points on a genus2 curve", Duke Math J. 90 (1997), 435463.

Schaefer, "2descent on the Jacobians of hyperelliptic curves", J. Number Theory 51 (1995), no. 2, 219232.
The part most directly relevant to the project is probably pp. 443452 of (1).
