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**Sample text**

However, x(P ) is an integer, and none of 3, 6, 11 or 38 is a perfect cube. e. the only torsion point is O). 7. Let p ≥ 2 be a prime number and let us define a curve Ep : y 2 = x3 + p2 . Since x3 + p2 = 0 does not have any rational roots, Ep (Q) does not contain points of order 2. Let P be a torsion point on Ep (Q). The list of all squares dividing 4A3 + 27B 2 = 27p4 is short, and by the Nagell-Lutz theorem the possible values for y(P ) are: y = ±1, ±p, ±p2 , ±3p, ±3p2 , and ± 3. 8. Thus, the torsion subgroup of Ep (Q) is isomorphic to Z/3Z, for any prime p ≥ 2.

1) We define ∆E , the discriminant of E, by ∆E = −16(4A3 + 27B 2 ). For a definition of the discriminant for more general Weierstrass equations, see for example [Sil86], p. 46. 4), and such that the discriminant of E is an integer. The minimal discriminant of E is the integer ∆E that attains the minimum of the set {|∆E | : E ∈ S}. In other words, the minimal discriminant is the smallest integral discriminant (in absolute value) of an elliptic curve that is isomorphic to E over Q. If E is the model for E with minimal discriminant, we say that E is a minimal model for E.

Examples of each of the possible torsion subgroups over Q. 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. 4. Let Eb : y 2 + (1 − b)xy − by = x3 − bx2 with b ∈ Q and ∆(b, c) = b5 (b2 − 11b − 1) = 0. Then, the torsion subgroup of Eb (Q) contains a subgroup isomorphic to Z/5Z, and (0, 0) is a point of exact order 5. Conversely, if E : y 2 = 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 Eb , for some b ∈ Q.