Magma V2.21-9 Fri Mar 11 2016 10:34:46 on atkin [Seed = 531199494] Type ? for help. Type -D to quit. Loading startup file "/home/samir/.magmarc" We will be doing computations with certain modular curves over the field Number Field with defining polynomial \$.1^3 - 13*\$.1^2 + 26*\$.1 - 13 over the Rational Field This is the unique cubic subfield of Q(zeta_{13}) 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 K^prime 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 K^prime are [ (2 : 2 : 1), (9 : -33 : 1) ] These have j-invariants [ 16581375, -3375 ] The modular curve X_0(11) is the elliptic curve Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field with Cremona label 11a1 Points on X_0(11) over K^prime are [ (0 : 1 : 0), (5 : -6 : 1), (16 : 60 : 1), (16 : -61 : 1), (5 : 5 : 1) ] The cusps of X_0(11) are [ Place at (16 : 60 : 1), Place at (0 : 1 : 0) ] The non-cuspidal points on X_0(11) over K^prime are [ (5 : -6 : 1), (16 : -61 : 1), (5 : 5 : 1) ] These have j-invariants [ -32768, -121, -24729001 ] Now we check the claim made in the proof that every point on X_0(20) over Q(zeta_{13})^+ is in fact defined over Q(sqrt{13}) The modular curve X=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 K^prime are [ (0 : 1 : 0), (4 : 10 : 1), (0 : 2 : 1), (-1 : 0 : 1), (0 : -2 : 1), (4 : -10 : 1) ] The quadratic twist of E by 13 is Elliptic Curve defined by y^2 = x^3 + 13*x^2 + 676*x + 8788 over Kp The Mordell--Weil group of this twist over K^prime is Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 The generators are (-13 : 0 : 1) (39 : -338 : 1) Thus if P belongs to X(K) then P+P^sigma and P-P^sigma both belong to X(Q(sqrt{13})) Thus 2P belongs to X(Q(sqrt{13}}) As the extension K/Q(sqrt{13}) is cubic, and the equation 2P=R has four solutions, it follows that 2P=2Q where Q belongs to X(Q(sqrt{13})) To see that P belongs to X(Q(sqrt{13})), it is enough to observe that X has full 2-torsion already over the rationals. Total time: 13.929 seconds, Total memory usage: 56.06MB