1203.sagews

def ker(E, ell1, ell2):
    P, Q, R = E.gens()
    E1 = E.change_ring(GF(ell1))
    E2 = E.change_ring(GF(ell2))
    K = []
    for a, b, c in GF(3)^3:
        S = ZZ(a)*P + ZZ(b)*Q + ZZ(c)*R
        S1 = E1(S)
        if len(S1.division_points(3)) > 0:
            S2 = E2(S)
            if len(S2.division_points(3)) > 0:
                K.append(S)
    return K

E = EllipticCurve('5077a')

ker(E, 23, 29)

[(0 : 1 : 0), (7/9 : -44/27 : 1), (1/4 : -21/8 : 1)]

P = ker(E, 23, 29)[-1]; P

(1/4 : -21/8 : 1)

E.change_ring(GF(11))(P).division_points(3)

[]