Specialization at T = 5/14 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -43445529415244318917275T^8 + 422168252009965455353400T^7 - 1000615470852214740048840T^6 - 287665459520844013324560T^5 + 730836999165531608469360T^4 - 1534018519968751162419840T^3 - 860145796269947075014080T^2 + 1257969101915425406539680T - 287979670569444598886832,
A6 = 3481769198723618054364520888653750T^12 - 51546572699746324664281711293253800T^11 + 257425214170732002290334218843166696T^10 - 384910798826712681438480236614322160T^9 - 547380365950409455224187028746997760T^8 + 1186458230569584965410644869136438720T^7 + 1903926669733832318282196311637144960T^6 - 342699085864060966883864121127625952T^5 - 1605189770963765735095620014272121520T^4 + 273990330917492951575782706024802880T^3 + 722163883191450327543709222733194560T^2 - 391773240304041316040915337560972160T + 59143024921117996675555945526480256.
It is a 2-neighbour of the fibration given in curve#4205 (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.
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