The conductor of these objects is an integer defined using primes when they do not have a good reduction. These objects are isomorphic over C if and only if they have the same j-invariant. The order of the zero of these objects’ Hasse-Weil L-functions at 1 is conjectured to equal their rank. A line drawn between two points on one of these objects must have a third intersection point, (15[1])assuming a point at (*) infinity is added. The SIKE system is based on isogenies of supersingular types of these objects, which are also used by Lenstra’s factorization algorithm. A correspondence between modular forms and these objects is given by the Taniyama-Shimura conjecture. (10[1]-5[1])These objects can be represented by real solutions to y squared equals x cubed plus a x plus b. For 10 points, name these objects used by Andrew Wiles in his proof of Fermat’s last theorem. ■END■ (0[1])