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Alternatively, you can also start to iterate the points on the curve without having to construct the whole list:
sage: k = GF(next_prime(7^5))
sage: E = EllipticCurve(k,[3,4])
sage: it = iter(E)
sage: it.next()
(0 : 1 : 0)
sage: it.next()
(0 : 2 : 1)
sage: it.next()
(0 : 16809 : 1)
sage: it.next()
(4 : 5217 : 1)
sage: it.next()
(4 : 11594 : 1)
sage: it.next()
(5 : 12 : 1)
What takes effort is to build the generators of the group. Once you have them, you just have to consider linear combinations. So you can also do
sage: gens = E.gens()
sage: gens
[(11860 : 14157 : 1), (15856 : 8344 : 1)]
sage: 12*gens[0] + 17*gens[1]
(14914 : 10227 : 1)
sage: 1243*gens[0] + 2*gens[1]
(7777 : 15736 : 1)
