curve #4216

y² + xy = x³ − 417002913150298113327185458283597038x + 106033586372454694396032803004873116035076891428674692

a-invariants: [1, 0, 0, -417002913150298113327185458283597038, 106033586372454694396032803004873116035076891428674692]
c-invariants: [20016139831214309439704901997612657825, -91613018625800855988196551543031836413863787190793920625]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 41404912517380058053828849521248900062672758440465027805081569107175800922832702586050
discriminant (Δ): -216181568530517366146285615548514438091945952930159580565362987329311859357695094068512883186262749184000000
Faltings height: 19.3774
naive height: 257.7143
primes of bad reduction: 2, 3, 5, 17, 37, 47, 379, 11243, 59311506215167649, 83590483637079143, 441972143131711413135924077034107198791
regulator: 8351713420605395981457695.6609688787137089192372765318058739058321394487147395015492810
submitted by: apkorte
submitted at: 2026-10-14 04:24:43 UTC
last updated: 2026-10-14 04:24:43 UTC

Witness: 24 independent points log in to add more points →

(452909821515327932, -100364551473740612072698366)
(26893074564522932, 307958675809150679854246634)
(309439289002661852, -81401109775008669698336926)
(80294065915455932, 270311660496309622594309634)
(9937749701269299044/49, 59213789518767364987517544590/343)
(370615051910158232, 48908608682303413766437634)
(38101740400185602, 300333753578466124472544824)
(385154128915423952, 50580453738211205905700174)
(270313833212840732, 114296305974776688909049634)
(71343451738511991932, 602578245546381469636709485634)
(20548189539726374828/49, 24029729105315787318148425902/343)
(-93259647989766244, -379620887616705241470199870)
(-9509914459154548, 331660049367939551387058674)
(-143902333740438148, 403808576323412947296843074)
(1443478023679801532, -1584856913736835486160837566)
(99215169658399723868/169, -553884995845824173465006910742/2197)
(17182537567242344357/16, 60605247067871861396494082261/64)
(-244228361978495272, 439670234877708379866953510)
(611613136962852782, -282445813822635639472551316)
(4711228508374275452, 10134610496279983145363095874)
(429721730473914716, -78683649615143594998602046)
(308687240288603822, -82001461952607999156661786)
(89383157298000032, 263580467012109448980223934)
(448575820244498372, 96118201279735576347151514)

Commentary

Specialization at T = -281/849 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -123856789139706169159947T^8 - 444860715403915335809784T^7 + 386197275608827081416504T^6 + 1527252937315431153244272T^5 - 1375436505233375995520880T^4 + 1122976665942695182127232T^3 - 920913248897305236746496T^2 - 558254638916475632295264T - 66664274811511860464112, A6 = 16359416402689331569871278519529286T^12 + 84307064804680497817942655822018088T^11 + 43582982500411768342190410573462056T^10 - 406313301062273227318175674455353520T^9 - 696574825662812869066591591106450400T^8 + 831815699985675007793533243030026816T^7 + 2339654226324227927870456252492273088T^6 - 2418192605459673378876521710121559264T^5 - 588423094738870081007533655250779760T^4 + 462827873819015983743769059572260800T^3 + 312326563016748800922506015981603904T^2 + 77162055820545612135770412695494272T + 6735862148389763480692013144006784. 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 04:24:43 UTC · history