curve #4218

y² + xy = x³ − 65968982729336490438114380739836780x + 5280964300161451687478424295351622815593133679755152

a-invariants: [1, 0, 0, -65968982729336490438114380739836780, 5280964300161451687478424295351622815593133679755152]
c-invariants: [3166511171008151541029490275512165441, -4562753155339494262731125347696029424216702912576699489]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 90265477399337853384281373437352459166657125981023179406109763937548861121056230
discriminant (Δ): 6325946571202494455758740084079538261526315235374602315702496053752203480506518093665022392975250944000000
Faltings height: 19.0176
naive height: 252.1371
primes of bad reduction: 2, 3, 5, 7, 11, 29, 53, 109, 167, 14037829051847, 4658169625016392513277, 21358837834525765320169690094860937
regulator: 26785346404809403135601.7172496427841537805677004776612063338134973280234995788247555
submitted by: apkorte
submitted at: 2026-10-14 05:25:04 UTC
last updated: 2026-10-14 05:25:04 UTC

Witness: 23 independent points log in to add more points →

(-227692964516182856, 92179736983824910309241428)
(210735424378337014, -27158801750052234432693062)
(51396726076626394, 45012724421917861284508678)
(446574852116491744, -254717286215107117300988972)
(-185639902584213656, 105498271685086370839356628)
(81797942134056304, 20787962635589763257571388)
(33179861933806744, -55934339904765336808148972)
(71893505586930904, 30163174848254404716673108)
(275161996300016344, -89232600030245633170652972)
(1077810822099415252, -1089149222036972088947005052)
(81164740474867744, 21477875172335060595222028)
(-76274483635438856, -99342664224873497185482572)
(182487974337689025496/2209, -2063389481272616072667601454356/103823)
(-83914743504716276, 101122863151977211348759108)
(91573077322307044, 2807017936036955966461828)
(79893193567205944, -22813234063956296936608172)
(-277970085393556232, 46263928612344660614534836)
(88193621474212564, -12202496788935187779532412)
(922334624194741144, -853853981645091004475622572)
(-63367348694353856, 95952057105467179804616428)
(42492739694793544, 50541940771079651629980628)
(-290117537441740556, -1027813001702662062850172)
(2361689947621851736/49, 16136945297702013978849463564/343)

Commentary

Specialization at T = -973/82 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -10490924381115083744475T^8 - 198606534222460688122680T^7 - 1065417976804438457255016T^6 + 1639325103621379811930736T^5 - 2065857830370934339051440T^4 + 1624515338975555277541440T^3 + 627292956712495652706816T^2 - 579571543143419272734816T - 175069545700351062329712, A6 = 377475523351270352670804404613750T^12 + 8093831427752091624508772362041000T^11 + 59608190005431403688311326659353800T^10 + 258877237945365511005499173975059280T^9 + 1166329210851912140109373033476192096T^8 - 3290737074386209591354797503931856128T^7 + 1076110014915408676877000781319873728T^6 + 1655411108858774108267005356846449184T^5 - 323687792595432052178005121163318384T^4 - 745457216313233319000016951349121600T^3 + 632802115601259765911745996693312T^2 + 143229663791151165669809219876399232T + 28115415195462890182082065162444416. It is a 2-neighbour of another fibration built here (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 2026-10-14 05:25:04 UTC · history