Magma V2.21-9 Fri Mar 11 2016 08:17:38 on lehner [Seed = 647087714] Type ? for help. Type -D to quit. Loading startup file "/home/samir/.magmarc" The modular curve X_0(20) is the elliptic curve Elliptic Curve defined by y^2 = x^3 + x^2 + 4*x + 4 over Rational Field with Cremona label 20a1 Points on X_0(20) over Q(zeta_{11})^+ are [ (0 : 1 : 0), (4 : 10 : 1), (0 : 2 : 1), (-1 : 0 : 1), (0 : -2 : 1), (4 : -10 : 1) ] The cusps of X_0(20) are [ Place at (4 : 10 : 1), Place at (4 : -10 : 1), Place at (-1 : 0 : 1), Place at (0 : -2 : 1), Place at (0 : 2 : 1), Place at (0 : 1 : 0) ] Thus every Q(zeta_{11})^+ point on X_0(20) is a cusp. The modular curve X_0(14) is the elliptic curve Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field with Cremona label 14a1 Points on X_0(14) over Q(zeta_{11})^+ are [ (0 : 1 : 0), (9 : 23 : 1), (2 : 2 : 1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1) ] The cusps of X_0(14) are [ Place at (9 : 23 : 1), Place at (2 : -5 : 1), Place at (1 : -1 : 1), Place at (0 : 1 : 0) ] The non-cuspidal points on X_0(14) over Q(zeta_{11})^+ are [ (2 : 2 : 1), (9 : -33 : 1) ] These have j-invariants [ 16581375, -3375 ] The modular curve X_0(19) is the elliptic curve Elliptic Curve defined by y^2 + y = x^3 + x^2 - 9*x - 15 over Rational Field with Cremona label 19a1 WARNING Class group unconditional proof is expensive. Hint: to obtain class group proven under GRH, input > SetClassGroupBounds("GRH"); before starting the current calculation. Points on X_0(19) over Q(zeta_{11})^+ are [ (0 : 1 : 0), (5 : -10 : 1), (5 : 9 : 1) ] The cusps of X_0(19) are [ Place at (5 : 9 : 1), Place at (0 : 1 : 0) ] The non-cuspidal points on X_0(19) over Q(zeta_{11})^+ are [ (5 : -10 : 1) ] These have j-invariants [ -884736 ] Total time: 1082.170 seconds, Total memory usage: 98.22MB