Tamagawa number divisibility of central<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-values of twists of the Fermat elliptic curve

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y =N. We first study 3-adic valuation of algebraic part value Hasse–Weil L-function L(C ,s) over ℚ at s=1, and exhibit a relation between 3-part its Tate–Shafarevich group number distinct divisors which are inert in imaginary quadratic field K=ℚ(-3). In case where ,1)≠0 is product split primes K, show that order as predicted conjecture Birch Swinnerton-Dyer perfect square.