Specialization at T = -71/67 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -19381964681583531802332T^8 - 28905723289750482456822T^7 + 237762949204059961327281T^6 + 531898331411046911194404T^5 - 2081641537694219583469905T^4 + 2900998586783373204925386T^3 - 3510952046475025537069569T^2 - 2402026805831481679416408T - 336697228013419606811427,
A6 = 1037857170981138962547584146233744T^12 + 2390096821086188960269294192921236T^11 - 17622524818710876412253124148231989T^10 - 68498401439427972797592193690386720T^9 + 142403571107393724072842085683710290T^8 + 537652195534688333016592840720348080T^7 + 242869176165494440760869297647114570T^6 - 5350043176257097379858948426236938120T^5 + 4432615889801018420373294120358506840T^4 + 3400474115253102291019214783235467100T^3 + 524125620210135846436694485054404435T^2 + 22638281992604199603670162474986120T + 4060904829303617481967199289389790.
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.
apkorte ·