Specialization at T = 113/69 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -36741768965072427329307T^8 - 175652477872961149894896T^7 + 69292619378708331563604T^6 + 676777688422838786893536T^5 - 656364464021699831415330T^4 + 2063898400836725040248640T^3 - 1358206219455103158319372T^2 - 1679620299616449035353360T - 380387730244665052932907,
A6 = 2972887747061499967014816310768950T^12 + 24971701851338630896585499476986576T^11 + 40958623108017859037060407448604996T^10 - 185504583684287485388498717737301760T^9 - 545461524648606440857833405079456086T^8 + 682919486625522225231856567512795600T^7 + 2337615182727970756165249196632581528T^6 - 2526741785379638456321704985966591088T^5 - 1756453725043689246538106067241758246T^4 + 628926877625456418301614685250524320T^3 + 1296733620825853845986136477111316036T^2 + 591132703001133261198186526777781680T + 89026717350891792884482981951773094.
It is a 2-neighbour of the fibration given in curve#4187 (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 ·