A. Student

Math 124 Problem Set 7

1. D=-155 There are four elements: [[1, 1, 39], [3, −1, 13], [3, 1, 13], [5, 5, 9]]. By the structure theorem, C_{−155} is isomorphic to either C_{2}xC_{2} or C_{4}. It is easy to verify that [1, 1, 39] is the identity. From this we find that [3, −1, 13] has order 4, so it must be that C_{−155} ⋍ C_{4}. D=-231 There are twelve elements: [1, 1, 58], [2, −1, 29], [2, 1, 29], [3, 3, 20], [4, −3, 15], [4, 3, 15], [5, −3, 12], [5, 3, 12], [6, −3, 10], [6, 3, 10], [7, 7, 10], [8, 5, 8]. Therefore C_{−231} ≃ C_{12} or C_{2}xC_{6}. The identity is [1, 1, 58]. Both [2, −1, 29] and [2, 1, 29] have order 6, which is impossible in C_{12}, so C_{−231} ≃ C_{2}xC_{6}. D=-660 There are eight elements: [1, 0, 165], [10, 10, 19], [11, 0, 15], [13, 4, 13], [2, 2, 83], [3, 0, 55], [5, 0, 33], [6, 6, 29]. The first element is the identity, and all others have order 2. Therefore C_{−660} ≃ C_{2}xC_{2}xC_{2}. D=-12104 There are forty-eight elements: (listed in an email from Professor Stein). By the structure theorem, C_{D} ≃ C_{48}, C_{4}xC_{12}, or C_{2}xC_{24}. The identity element is [1, 0, 3026], and using it we find two elements of order four: [45, −26, 71] and [50, −36, 67], eliminating everything but C_{4}xC_{12}. D=-10015 There are fifty-four elements (listed in an email from Professor Stein). Therefore C_{D} ≃ C_{3}xC_{18} or C_{54}. The identity is [1, 1, 2504]; from this we find two elements with order 9: [10, −5, 251] and [10, 5, 251]. Therefore the group cannot be C_{54}, so C_{D} ≃ C_{3}xC_{18}.

2. The three graphs are on the next page, plotted in MAPLE.

3. Differentiating implicitly, the slope of the tangent at (x, y) is \frac{3x^{2}}{2y}. At (3, 5), the slope is \frac{27}{10}, and the tangent line has equation y = \frac{27x−31}{10}. Substituting into the relation y^{2} − x^{3} = −2, we have (\frac{27x−31}{10})^{2} − x^{3} = −2, which simplifies to the polynomial

100x^{3} − 729x^{2} + 1674x − 1161 = 0.

This polynomial has a double root at x = 3, so it factors into (x − 3)^{2}(100x − 129), giving a rational root with x = 1.29. Therefore (1.29, .383) is a rational solution to the original equation.