On the analytic rank of the twin prime elliptic curve y²=x(x-2)(x-p)

Let p≥ 7 and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve Eₚ:y²=x(x-2)(x-p) was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of Eₚ/Q. I show that for p≡ 3,5 8, the analytic rank of Eₚ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of Eₚ for all twin prime pairs. I also show that Beers' conjecture, that for p≡ 1 8 the rank of Eₚ is two, is false as stated because E₇₃ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if p≡ 1 8 then the rank of Eₚ is zero or two (Conjecture 5.3.1).