Analogue of the Brauer-Siegel theorem for Legendre elliptic curves

We prove an analogue of the Brauer-Siegel theorem for the Legendre elliptic curves over K=Fq(t). Namely, denoting by Ed the elliptic curve with model y2=x(x+1)(x+td) over K, we show that, for d→∞ ranging over the integers, one haslog⁡(|СХ(Ed/K)|⋅Reg(Ed/K))∼log⁡H(Ed/K)∼log⁡q2⋅d. Here, H(Ed/K) denotes the exponential differential height of Ed, СХ(Ed/K) its Tate-Shafarevich group (which is known to be finite), and Reg(Ed/K) its Néron-Tate regulator.