Specialization at T = 4/19 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -62238598979262328847367T^8 + 114310404580625317891332T^7 + 345964098754055988215811T^6 - 432486848349735134109966T^5 + 175844415801606548060475T^4 - 539957014758046237778784T^3 - 3466666840492299593283363T^2 + 1375801562449141429517514T - 457847843272340951907252,
A6 = 6231113960947468288851293055208170T^12 - 17986527908278999930497783451836564T^11 - 63510145446922339309436221943522517T^10 + 196072827486310127387111443914446460T^9 + 329896017984738220842162864843822384T^8 - 977730912392624912334102291243654336T^7 - 1090216443601797579432503967842970390T^6 + 2401580225827262855577615397568975952T^5 + 1437988402318580839377591372944875134T^4 - 1577869275593930979134666639990964780T^3 + 1723891172774355272317470092631767475T^2 - 538001952495578595025088868536038092T + 119515751079584041083046899484291104.
It is a 2-neighbour of the fibration given in curve#4187 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.
apkorte ·