Patrick Reichert www.patrick-reichert.de E = Elliptische Kurve X_1(n) in den Variablen x, y und Parameter c und Basispunkt OO P = n-Torsionspunkt (0,0) auf E f = Weil-Funktion E -> C mit Divisor div(f)=n[P] - n[OO] n = 4: E = EllipticCurve([1, c, c, 0, 0]); f = x^2 - y; n = 5: E = EllipticCurve([c+1, c, c, 0, 0]); f = -x^2 + x*y + y; n = 6: E = EllipticCurve([1-c, -c*(c+1), -c*(c+1), 0, 0]); f = y^2 - (c + 1)*x*y - (c + 1)^2*y + (c + 1)^2*x^2; n = 7: E = EllipticCurve([1-c-c^2, c^2*(c+1), c^2*(c+1), 0, 0]); f = (c - 1)*y^2 + x^2*y - c^3*x*y + c^4*y - c^4*x^2; n = 8: E = EllipticCurve([1-2*c^2, -c*(2*c+1)*(c+1)^2, -c*(2*c+1)*(c+1)^3, 0, 0]); f = x*y^2 + (2*c + 3)*(c + 1)^3*y^2 - 2*(c + 1)^2*x^2*y - (2*c + 1)^2*(c + 1)^4*x*y - (2*c + 1)^2*(c + 1)^7*y + (2*c + 1)^2*(c + 1)^6*x^2; n = 9: E = EllipticCurve([c^3 + c^2 + 1, c^2 * (c+1) * (c^2+c+1), c^2 * (c+1) * (c^2+c+1), 0, 0]); f = y^3 + (c - 1)*(c^2 + c + 1)*x*y^2 + (c^2 + c + 1)^2*(c^3 + 2*c - 1)*y^2 - (2*c - 1)*(c^2 + c + 1)^2*x^2*y + c^4*(c^2 + c + 1)^3*x*y + c^4*(c^2 + c + 1)^4*y - c^4*(c^2 + c + 1)^4*x^2; n = 10: E = EllipticCurve([-c^3 - 2*c^2 + 4*c + 4, (c+1) * (c+2) * c^3, (c+1) * (c+2) * (c^2+6*c+4) * c^3, 0, 0]); f = 2*(c^2 - 2*c - 2)*y^3 + x^2*y^2 - (2*c + 1)*c^4*x*y^2 + (c^3 + 16*c^2 + 22*c + 8)*c^6*y^2 - (3*c + 2)*c^6*x^2*y + (c + 1)^2*c^10*x*y - (c + 1)^2*(c^2 + 6*c + 4)*c^12*y + (c + 1)^2*c^12*x^2; n = 12: E = EllipticCurve([6*c^4 - 8*c^3 + 2*c^2 + 2*c - 1, -c * (c-1)^2 * (2*c-1) * (2*c^2-2*c+1) * (3*c^2-3*c+1), -c * (c-1)^5 * (2*c-1) * (2*c^2-2*c+1) * (3*c^2-3*c+1), 0, 0]); f = y^4 + (6*c^2 - 8*c + 3)*(2*c^2 - 2*c + 1)*x*y^3 + (c - 1)^4*(2*c^2 - 2*c + 1)^2*(36*c^4 - 90*c^3 + 96*c^2 - 49*c + 10)*y^3 + 3*(c - 1)^2*(5*c^2 - 6*c + 2)*(2*c^2 - 2*c + 1)^2*x^2*y^2 + (c - 1)^4*(14*c^2 - 16*c + 5)*(3*c^2 - 3*c + 1)^2*(2*c^2 - 2*c + 1)^3*x*y^2 + (c - 1)^8*(3*c^2 - 3*c + 1)^2*(2*c^2 - 2*c + 1)^4*(12*c^4 - 42*c^3 + 57*c^2 - 33*c + 7)*y^2 + 2*(c - 1)^6*(9*c^2 - 10*c + 3)*(3*c^2 - 3*c + 1)^2*(2*c^2 - 2*c + 1)^4*x^2*y + (2*c - 1)^2*(c - 1)^8*(3*c^2 - 3*c + 1)^4*(2*c^2 - 2*c + 1)^5*x*y - (2*c - 1)^2*(c - 1)^13*(3*c^2 - 3*c + 1)^4*(2*c^2 - 2*c + 1)^6*y + (2*c - 1)^2*(c - 1)^10*(3*c^2 - 3*c + 1)^4*(2*c^2 - 2*c + 1)^6*x^2;