Without explicit numbers, what tools are available to prove the rank of an Elliptic curve is at least 1?
To provide a concrete example to aid discussion, consider the curves: $$ y^2 + y = x^3 - 18(k+1)$$ with the constraint on $k$ that $p = 72k + 71$ is a prime.
I have looked through many such curves for small $k$ (first 1000 meeting that constraint) and the rank (or analytic rank if Sage has trouble calculating the basis explicitly) is always odd.
How could we prove (or disprove) that this is true for all such $p$?
The above was chosen so that the discriminant has a known factorization: $$\Delta = -2^4 3^3 p^2 $$ While not in Weierstrass form, in case it is easier to approach with elementary techniques, note that the curve is birationally equivalent to: $$y^2 = 4x^3 - p$$