Specialization at T = 7/1 of a rank-17 fibration of X1092 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -179953294423718847382027T^8 - 2422012411772909548445296T^7 - 8684786346925804611549268T^6 - 4573475462542070744244864T^5 - 8326574805923000528641650T^4 + 35444345428350105767505072T^3 + 17731786323723333634379340T^2 - 19763869255646989140411552T - 10177374299919296571719307,
A6 = 29433046587806298151568576053541446T^12 + 603520435264734361699466145077660112T^11 + 4382581162253179659711002005983168540T^10 + 11396517244561787350481309183444475360T^9 - 10659015250764560811790628626096653654T^8 - 80586644259246975736139435374429063488T^7 + 42318930188195229744508377014871751368T^6 + 137262458200367432785871414671050119040T^5 - 23161725646205355538786516956002840214T^4 - 116199202372193956963687253523228736080T^3 - 14693145837274105663456587481964821764T^2 + 37328369442971825728801730381692902624T + 12558368421895894107424021098828811206.
It is a 2-neighbour of the fibration given in curve#728: 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.
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