Symbolic points in an elliptic curve over $\mathbb{Q}$ of the form $(u/e^{2},v/e^{3})$

Question

Suppose I have an elliptic curve $E : y^{2} = x^{3} + D$ over $\mathbb{Q}$ , where $D$ is a symbolic constant (an integer). I want to define two points in the curve of the type $P = (u_{1}/e_{1}^{2}, v_{1}/e_{1}^3)$ and $Q = (u_{2}/e_{2}^{2},v_{2}/e_{2}^{3})$ where $u_{i},v_{i},e_{i}$ are symbolic constants (integers). I don't know how to define them in MAGMA, I tried doing so by defining the curve in some function field, but it failed. How to achieve this representation using MAGMA? Basically, I want to see the representation of the point $P+Q$ in terms of $u_{i},v_{i},e_{i}$,

Answer 1

Given an elliptic curve $$ y^2 = x^3 + ax + b $$ and points on the curve $\,P = (u_1/e_1^2, v_1/e_1^3),\,$ $\,Q = (u_2/e_2^2,v_2/e_2^3),\,$ then $\,P+Q = (u_3/e_3^3, v_3/e_3^3)\, $ where $$u_3 = e_2^2 e_1^4 u_1 u_2^2+e_2^4 e_1^2 u_1^2 u_2-e_2^6 u_1^3+e_1^6 v_2^2-2 e_2^3 e_1^3 v_1 v_2+e_2^6 v_1^2 -e_1^6u_2^3, $$ $$v_3 = e_1^9 u_2^3 v_2 - 2 e_2^3 e_1^6 u_2^3 v_1 - 3 e_2^4 e_1^5 u_1^2 u_2 v_2 + 3e_2^5 e_1^4 u_1 u_2^2 v_1 + 2 e_2^6 e_1^3 u_1^3 v_2-e_2^9 u_1^3 v_1 - e_1^9v_2^3 + 3 e_2^3 e_1^6 v_1 v_2^2 - 3e_2^6 e_1^3 v_1^2 v_2 + e_2^9 v_1^3, $$ $$e_3 = e_1 e_2 (e_1^2 u_2-e_2^2 u_1).$$

If $\,P = (u/e^2, v/e^3),\,$ then $\,2P = (u_2/e_2^2,v_2/e_2^3),\,$ where $$ u_2 = a^2 e^8+6 a e^4 u^2+9 u^4-8 u v^2,\qquad e_2 = 2 e v, $$ $$ v_2 = -a^3 e^{12}-9 a^2 e^8 u^2-27 a e^4 u^4+12 a e^4 u v^2-27 u^6+36 u^3 v^2-8 v^4. $$ I found the formulas I needed on a web page at Project Nayuki.

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As a check, to add two points on the curve $\,P = (x_1, y_1),\,$ $\,Q = (x_2, y_2),\,$ then $\, P+Q = (x_3, y_3)\, $ where $\,m := (y_1-y_2)/(x_1-x_2),\,$ $\,x_3 = -x_1-x_2,\,$ and $\,y_3 = -((x_1y_2-x_2y_1)/(x_1-x_2) + x_3m).$ This should agree with the previous formulas using $\,(u_3,v_3).$