Specialization at T = 3/7 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where
A4 = -56986316681380998912T^8 + 24457666340476410624T^7 + 535487594425392365712T^6 + 48800690383254862104T^5 - 361470535143673852503T^4 - 1833035186758586060700T^3 - 8027564283553513223079T^2 - 1856193563269433892114T - 193818285257580953532,
A6 = 176927718245814623549804322816T^12 - 116292625590383842700445954048T^11 - 3367104967138625608924904544000T^10 + 2020152346629037937923069338624T^9 + 29203681898463675958459856569392T^8 - 13720510267525899173180962213992T^7 - 142330533991632689827729911017922T^6 + 21328459230505455903018021874128T^5 + 318228745035755178053554583805228T^4 + 157407522251338495321687387801980T^3 + 99390588041592331039607736906018T^2 + 19564837728780267552757168309932T + 952356233459948553164561681844.
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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