Computational Results on the Congruent Number Family: E_n : y^2 = x^3 - n^2 x

We present computational results on the elliptic curve family E_n : y^2 = x^3 - n^2 x corresponding to the classical congruent number problem. Using PARI/GP (cypari2) for rank computation and Singular for algebraic geometry, we verify a rank pattern with no counterexamples in thousands of cases.

Key results include: (1) A "never-rank-2 rule" verified for n ≤ 10000 with zero failures; (2) Cremona table consistency of 99.75%; (3) Parity and vacuity analysis of the Tunnell criterion.