Introduction to Abstract Algebra
Introduction to Abstract Algebra MAT 334
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This 1 page Class Notes was uploaded by Dr. Taya Dickens on Wednesday October 28, 2015. The Class Notes belongs to MAT 334 at Utica College taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/230525/mat-334-utica-college in Mathematics (M) at Utica College.
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Date Created: 10/28/15
ll Classify the torsion subgroup of the rational points on the elliptic curve y2 zg212 7 31 The discriminant of the polynomial 13 212 7 31 is D 144 By the LutzNagell theorem if P z y is a rational points on the elliptic curve of nite order then either y 0 or y divides 144 Running through the diVisors of 144 we nd that the points of nite order are 0 0 l 0 73 0 71 2 71 72 3 6 and 3 76 So including the point at in nity the group of nite rational points on the elliptic curve 1s 0070707170773707717 27717 72 3767 37 76 This is a nite Abelian group of order 8 so it must be isomorphic to Z37Z4 69 Z27 0r Z2 69 Z2 69 Z2 We consider the points 00 10 and 730 These three points lie on the zaxis and the tangent lines to the curve at these points are all vertical Thus the third point of intersection is the point at in nity So the orders of each of these points is 2 So the group has at least three points of order two namely 00 l 0 and 73 0 The group Z3 has only 1 element of order two namely the element 4 since 4 04 So the group cannot be Zgi So we are left with G Z4 69 Z2 and H Z2 69 Z2 69 Z2i One main difference is that the group G has elements of order 4 whereas the group H has no elements of order 4 Consider the element 712 on the elliptic curve The tangent line to the curve at the point 71 2 intersects the curve at 10 which when re ected over the zaxis is 710 So 712 712 10i At any rate the point 71 0 is not a point of order 2 So the group cannot be Z2 69 Z2 69 Z2 Therefore the group is Z4 69 Z2i