Specialization at T = -65/21 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -42783457648311062145792T^8 + 356652061732127265219072T^7 - 776392718276863895491584T^6 - 594563894687761097733504T^5 + 1610134904768785320105720T^4 - 1299142963618345803173436T^3 - 1310847137770922715048759T^2 + 2293995846725083806581718T - 762637489081909275352167,
A6 = 3393863243684538700888419573325824T^12 - 42546028233387428372499985647230976T^11 + 184326642514085964348103427989426176T^10 - 202262449736636480735819511193036800T^9 - 605066230051264992235683353165205120T^8 + 1246062737155179656440992962298863808T^7 + 1477591518827520114475641040966728528T^6 - 1553487810870706704105291467455269168T^5 - 2067093448474470395926495237989292755T^4 + 1400402544246467407554684971457155760T^3 + 1384929818086858833031645353525974259T^2 - 1332090731990002399128362594486258406T + 294374686993228956276286908014904426.
It is a 2-neighbour of the fibration given in curve#4191 (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 ·