## Experimental Mathematics

- Experiment. Math.
- Volume 12, Number 3 (2003), 311-318.

### Numerical Investigations Related to the Derivatives of the *L*-Series of Certain Elliptic Curves

#### Abstract

In Zagier and Kramarz, the authors computed the critical value of the *L*-series of the family of elliptic curves {\small $x^3+y^3=m$} and they pointed out some numerical phenomena concerning the frequency of curves with a positive rank and the frequency of occurrences of the Tate-Shafarevich groups {\small$\TSg$} in the rank 0 case (assuming the Birch and Swinnerton-Dyer conjecture). In this paper, we give a similar study for the family of elliptic curves associated to simplest cubic fields. These curves have a nonzero rank and we discuss the density of curves of rank 3 that occurs. We also remark on a possible positive density of nontrivial Tate-Shafarevitch groups in the rank 1 case. Finally, we give examples of curves of rank 3 and 5 for which the group {\small $\TSg$} is nontrivial.

#### Article information

**Source**

Experiment. Math., Volume 12, Number 3 (2003), 311-318.

**Dates**

First available in Project Euclid: 15 June 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1087329234

**Mathematical Reviews number (MathSciNet)**

MR2034395

**Zentralblatt MATH identifier**

1083.11041

**Subjects**

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Secondary: 11Y35: Analytic computations 15A52

**Keywords**

Elliptic curves simplest cubic fields analytic rank Tate-Shafarevich group L-series

#### Citation

Delaunay, C.; Duquesne, S. Numerical Investigations Related to the Derivatives of the L -Series of Certain Elliptic Curves. Experiment. Math. 12 (2003), no. 3, 311--318. https://projecteuclid.org/euclid.em/1087329234