High rank elliptic curves with torsion group ℤ/(2ℤ)

Aguirre, Julián; Castañeda, Fernando; Peral, Juan

doi:10.1090/S0025-5718-03-01547-3

High rank elliptic curves with torsion group $\mathbb {Z}/(2\mathbb {Z})$
HTML articles powered by AMS MathViewer

by Julián Aguirre, Fernando Castañeda and Juan Carlos Peral PDF

Math. Comp. 73 (2004), 323-331 Request permission

Abstract:

We develop an algorithm for bounding the rank of elliptic curves in the family $y^2=x^3-B x$, all of them with torsion group $\mathbb {Z} /(2 \mathbb {Z})$ and modular invariant $j=1728$. We use it to look for curves of high rank in this family and present four such curves of rank $13$ and $22$ of rank $12$.

References

J. Aguirre, F. Castañeda, and J. C. Peral, High rank elliptic curves of the form $y^2=x^3+Bx$, Rev. Mat. Complut. 13 (2000), no. 1, 17–31. MR 1794901, DOI 10.5209/rev_{r}ema.2000.v13.n1.17088
J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
Stéfane Fermigier, Exemples de courbes elliptiques de grand rang sur $\textbf {Q}(t)$ et sur $\textbf {Q}$ possédant des points d’ordre $2$, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 10, 949–957 (French, with English and French summaries). MR 1393541
Stéfane Fermigier, Construction of high-rank elliptic curves over $\textbf {Q}$ and $\textbf {Q}(t)$ with non-trivial $2$-torsion (extended abstract), Algorithmic number theory (Talence, 1996) Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 115–120. MR 1446503, DOI 10.1007/3-540-61581-4_{4}6
Stéfane Fermigier, Une courbe elliptique définie sur $\mathbf Q$ de rang $\geq 22$, Acta Arith. 82 (1997), no. 4, 359–363 (French). MR 1483688, DOI 10.4064/aa-82-4-359-363
Koh-ichi Nagao, On the rank of elliptic curve $y^2=x^3-kx$, Kobe J. Math. 11 (1994), no. 2, 205–210. MR 1329432
Nicholas F. Rogers, Rank computations for the congruent number elliptic curves, Experiment. Math. 9 (2000), no. 4, 591–594. MR 1806294
Tetsuji Shioda, Construction of elliptic curves with high rank via the invariants of the Weyl groups, J. Math. Soc. Japan 43 (1991), no. 4, 673–719. MR 1126145, DOI 10.2969/jmsj/04340673
Tetsuji Shioda and Hisashi Usui, Fundamental invariants of Weyl groups and excellent families of elliptic curves, Comment. Math. Univ. St. Paul. 41 (1992), no. 2, 169–217. MR 1190156
Joseph H. Silverman and John Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1171452, DOI 10.1007/978-1-4757-4252-7
Tate, J., “Rational points on elliptic curves”, Phillips Lectures, Haverford College, 1961.