curve #2177

y² = x³ + x² − 97870839201297877642320x − 11333666432314542438509577862823856

a-invariants: [0, 1, 0, -97870839201297877642320, -11333666432314542438509577862823856]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 12853118643121903611159945035499392612472 ☆ record for ℤ/2ℤ torsion, rank ≥ 14
discriminant (Δ): 4507171627743587173924638680560347278469725599572894841534205042568192
Faltings height: 12.1289
naive height: 170.4274
primes of bad reduction: 2, 3, 7, 11, 17, 29, 59, 71, 73, 79, 109, 163, 13306129, 45371593, 54443929
regulator: 62973477324645.4667993663433724695116089596544533008066734420302153
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-26 15:39:31 UTC
last updated: 2026-09-26 15:39:31 UTC

Witness: 14 independent points log in to add more points →

(-3353529957512485430082784750041005806975224/22205921555640147008636976708721, 164037877543143208360789926461314159140532132217130540049271844/104641318196827969442140680204345625119266421769)
(-11103128698519704231241320/60583448230369, 9974991224364110639683058040634135836/471553510888661635153)
(18857292698026818148671567/2058648170401, -81839387866953043016560529634676367316/2953746336243184399)
(3224233401627898303259198580681/5857264146115103296, 4518250331655898505340764933167183577639510619/14175624966110140782402054656)
(-165055707336, -17994380419260252)
(-48275246395677768/231361, 201927536050066945870116/111284641)
(-60478103113582128083597413477813676491870104/301357510244185052038830173710441, -78480298360912270979392936061410307978550426276263329375335179436/5231461441915905359092918638675632383358759374261)
(-609149588307/4, -50000425726980603/8)
(-5918169654404667982044971663694769164441/28939678633349239015741898041, 55845199822614563007193782056815174836028847227280403795288/4923121523637583211962561056188115443412589)
(105309370873523814325/147039876, 953424342786378843053389238887/1783005536376)
(3432794377664274399625294429145190243494470949109298001637419883/6669117400590246727264959865964659345910578641734809, -148820310535546463297190524911829889948149256355516484240814851645660329149181566883400154875502/544631233915293305763248663177517650862729288878229797900228847853788304157123)
(360996714104, 19490473505835612)
(-593813231674369021283403921065865172139743270083446365278005200727569860444124883185711270633229061134518536/3103109158765021414750156168589544857058818934468241490687424485391681236318994798581349708077569, -107610832843940120711561899384715407672467486863090435258614198527914023901216211892271462600717750175726987882888350622453423752414524295595756406272150556794972/5466326644020269145940291390757770610877900872159178037690488831856044063246540383650731393593384425542408557100807790886406893969488338092074753)
(-72912535631704834756826383863220928184/482925218280172111460097481, 12190275306377863522063201499854987554122820205426072596/10612551900693906942525005382766004975621)

Commentary

This was found searching in a Kihara family. More details will be forthcoming -- or contact the author by email for details.

last edited by Alvaro Lozano-Robledo at 2026-09-26 15:39:31 UTC · history