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curve #6127

y2 = x3 − x2 − 43418418425420867878323945025x + 3808109786561037988151394670153211463827077
a-invariants
[0, -1, 0, -43418418425420867878323945025, 3808109786561037988151394670153211463827077]
c-invariants
[2084084084420201658159549361216, -3290206855588724317258298473802425747450427264]
rank (lower bound)
≥ 22
torsion subgroup
trivial
conductor (N)
23354182517806638348389168280218084681450183982791012291203885349473113960
discriminant (Δ)
-1026294475881151602837448895107916439936527055998771574649983863078309552464536492000000
Faltings height
15.4328
naive height
209.6146
primes of bad reduction
2, 3, 5, 7, 13, 19, 49293430093, 521476916652319, 17728310468531826294756873440508535824823
regulator
1653133047492139489049.985581625998874570379505463744614697793993886419584473667315
submitted by
apkorte
submitted at
last updated

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Commentary

Specialization at T = 191/42 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -135905733791177100879168T^8 + 1170105215005157329310976T^7 - 3263768771352480615865392T^6 + 2346518634849035692218000T^5 + 245936114819649867667320T^4 - 1432517413140009535585824T^3 + 956236108008185762686704T^2 + 156236369911740531083592T - 257526183060059049073467, A6 = 20737642716792465943326120257995008T^12 - 244496554556977289037909247323669504T^11 + 1157988470380485097232050906009638528T^10 - 2790680074288331258566479141986856192T^9 + 3101966268250480126177717673389002384T^8 + 656755301509986568870676405232297792T^7 - 3435411333307752838327381281470537136T^6 + 658445704499369672945800020331508016T^5 + 1474214217968126497120054729756719768T^4 - 229935451360122136801662723346458528T^3 - 405994673197848668282333146509853248T^2 + 15664520646757240102474163329610664T + 58525848665960980471096021479565974. 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.

last edited by apkorte at · history

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