The Elliptic Curve Database for Conductors to 130000

Abstract

Abstract. Tabulating elliptic curves has been carried out since the earliest days of machine computation in number theory. After some historical remarks, we report on significant recent progress in enlarging the database of elliptic curves defined over Q to include all those of conductor N ≤ 130000. We also give various statistics, summarize the data, describe how it may be obtained and used, and mention some recent work regarding the verification of Manin's "c = 1" conjecture. Background and historyTabulating elliptic curves has been carried out since the earliest days of machine computation in number theory. In this article we concentrate on tables which claim to contain complete lists of elliptic curves with conductors in certain ranges. Other tables exist, notably tables of curves with prime conductor by Brumer and McGuinness [4] and, more recently, by Stein and Watkins [21].We first review the tables existing before 1990, and then describe the tables we have compiled since then, concentrating on the large increase in the data available since mid-2005. We will describe the origins of the tables and give some information on the methods used to compile them. We give a summary of the data obtained to date, describe how to obtain and use the data, and mention some recent work regarding the verification of Manin's "c = 1" conjecture.

ON THE DIOPHANTINE EQUATION x2 + C = 2yn

“…and the notation f 1 → 4224A1 means that f 1 corresponds to the elliptic curve 4224A1 in Cremona's tables [20].…”

Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

“…Clearly, the Birch and Swinnerton-Dyer conjecture holds if and only if #X(Q, E) an > 0 and, for each prime p, BSD(E/Q, p) is true. The rank conjecture has been verified for E/Q of conductor N < 130 000 [15]. This is possible because E is known to be modular and the analytic rank is at most three.…”

Almost powers in the Lucas sequence

“…Eliminating Small Exponents p. We will later use the modular approach to help us solve the equations L 2m = qy p by attaching Frey curves to one or more of the three auxiliary equations (13)- (15). In the modular approach, it is useful to avoid small exponents p where the Galois representation may be reducible.…”

“…(18) We attach a Frey curve to each solution of equations (13), (14), (15) and use them to help solve our problem. Experience shows that the simultaneous use of more than one Frey curve gives far better information than the use of just one Frey curve.…”

“…In each case the pair of newforms remaining will consist of two rational ones, and hence correspond to elliptic curves. We refer to these elliptic curves using their Cremona reference (as in Cremona's book [14], or his extended online tables [15]). Proposition 9.6.…”