Multi-Page Printing

2.5. The torsion subgroup 33 Curve Torsion Generators y2 = x3 − 2 trivial O y2 = x3 + 8 Z/2Z (−2,0) y2 = x3 + 4 Z/3Z (0,2) y2 = x3 + 4x Z/4Z (2,4) y2 − y = x3 − x2 Z/5Z (0,1) y2 = x3 + 1 Z/6Z (2,3) y2 = x3 − 43x + 166 Z/7Z (3,8) y2 + 7xy = x3 + 16x Z/8Z (−2,10) y2 + xy + y = x3 − x2 − 14x + 29 Z/9Z (3,1) y2 + xy = x3 − 45x + 81 Z/10Z (0,9) y2 + 43xy − 210y = x3 − 210x2 Z/12Z (0,210) y2 = x3 − 4x Z/2Z ⊕ Z/2Z ( (2,0) ) y2 = x3 + 2x2 − 3x Z/4Z ⊕ Z/2Z ((0,0)) (3,6) (0,0) y2 + 5xy − 6y = x3 − 3x2 Z/6Z ⊕ Z/2Z ( (−3,18) (2,−2) ) y2 + 17xy − 120y = x3 − 60x2 Z/8Z ⊕ Z/2Z ( (30,−90) (−40,400) ) Figure 4. Examples of each of the possible torsion subgroups over Q. Furthermore, it is known that, if G is any of the groups in Eq. 2.5, there are infinitely many elliptic curves whose torsion subgroup is isomorphic to G. See, for example, [Kub76], Table 3, p. 217. For the convenience of the reader, the table in Kubert’s article is reproduced in Appendix E. Example 2.5.4. Let Et : y2 + (1 − t)xy − ty = x3 − tx2 with t ∈ Q and Δt = t5(t2 − 11t − 1) = 0. Then, the torsion subgroup of Et(Q) contains a subgroup isomorphic to Z/5Z, and (0,0) is a point of exact order 5. Conversely, if E : y2 = x3 + Ax + B is an elliptic curve with torsion subgroup equal to Z/5Z, then there is an invertible change of variables that takes E to an equation of the form Et for some t ∈ Q. See also Examples E.1.1 and E.1.2. A useful and simple consequence of Mazur’s theorem is that if the order of a rational point P ∈ E(Q) is larger than 12, then P must be a point of infinite order and, therefore, E(Q) contains an infinite