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In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.
The paper proves that the Birch and Swinnerton-Dyer conjecture is false.
We generalize a construction of families of moderate rank elliptic curves over $mathbb{Q}$ to number fields $K/mathbb{Q}$. The construction, originally due to Steven J. Miller, Alvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silv
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
We give an elementary and self-contained introduction to pairings on elliptic curves over finite fields. For the first time in the literature, the three different definitions of the Weil pairing are stated correctly and proved to be equivalent using
A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which every cur