Specialization at T = 25/24 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -3261288267173019191472T^8 + 16336709243720609162688T^7 - 18268306752455049094272T^6 - 19274483100900643686432T^5 + 47527616284455427617672T^4 - 42492106937679957084528T^3 - 37117805963754630710128T^2 + 57264934594581910892472T - 17428523548140354643227,
A6 = 73778746260046465803130020065664T^12 - 557691636286597619819213618736384T^11 + 1241644089075433039709482561447680T^10 + 319292057952411507722665432506240T^9 - 3684776099503733525147843819041632T^8 - 105975540379555346975431938448896T^7 + 6959425978364243633608708638354240T^6 + 2168365187018207896801103646494784T^5 - 9811177379114913645781233251073240T^4 + 1286921179016092176554695571280336T^3 + 6464160388118625330104562158316256T^2 - 4394066368078192631786533039340328T + 883707771835909746830128944197430.
It is a 2-neighbour of the fibration given in curve#3589: 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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