Specialization t=-8/1 of the universal X_1(7) Tate normal form family y^2+(-t^2+t+1)xy+(-t^3+t^2)y=x^3+(-t^3+t^2)x^2 (K3 elliptic surface, rational 7-torsion point (0,0)).
Specialization t=-8/1 of the universal X_1(7) Tate normal form family y^2+(-t^2+t+1)xy+(-t^3+t^2)y=x^3+(-t^3+t^2)x^2 (K3 elliptic surface, rational 7-torsion point (0,0)). Team Euler (operator Laohuang).
Specialization t=-8/1 of the universal X_1(7) Tate normal form family (K3 elliptic surface) E_t: y^2+(-t^2+t+1)xy+(-t^3+t^2)y=x^3+(-t^3+t^2)x^2 carrying the rational 7-torsion point (0,0). Found by an exhaustive low-height scan of 10.9M rational specializations (|p|,|q|<=3000), ranked by global-minimal-model naive height log(max(|c4|^3,|c6|^2)) = 53.4649. Torsion Z/7Z verified with PARI/GP elltors on the minimal model; non-torsion witness point found by ellratpoints (B=10^6) and verified on-curve; root number W=-1, consistent with odd rank, so rank >= 1 also follows from parity. Self-discovered specialization, not a literature curve.
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