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On the arithmetic of a family of twisted constant elliptic curvesLet F-r be a finite field of characteristic p > 3. For any power q of p, consider the elliptic curve E = E-q,E-r defined by y(2) = x(3) + t(q) - t over K = F-r (t). We describe several arithmetic invariants of E such as the rank of its Mordell-Weil group E(K), the size of its Neron-Tate regulator Reg(E), and the order of its Tate-Shafarevich group III(E) (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of p modulo 6. For instance III(E) either has trivial p-part or is a p-group. On the other hand, we show that the product III(E) Reg(E) has size comparable to r(q/6) as q -> infinity, regardless of p (mod 6). Our approach relies on the BSD conjecture, an explicit expression for the L -function of E, and a geometric analysis of the Neron model of E.