curve #1836

y² + xy + y = x³ + x² − 62768067307229828411090203757990603067021720929984465938x + 189210859514269843332267105356636723832238980724137875777900357363399313479474032031

a-invariants: [1, 1, 1, -62768067307229828411090203757990603067021720929984465938, 189210859514269843332267105356636723832238980724137875777900357363399313479474032031]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1564270218869624840947724233813568092984047710847969648779404950 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 13
discriminant (Δ): 360990311538874708730997365171268606233029129752097169822769787517495790192554624276883003712410947887987280251880525619791211173015718662408838602545114607673600000000
Faltings height: 30.9587
naive height: 397.0507
primes of bad reduction: 2, 3, 5, 11, 19, 31, 43, 103, 113, 167, 307, 317, 439, 479, 503, 823, 11831, 13151, 22573, 24469, 26452134174795739
regulator: 8630366097628887.087877497253389884463291775311562944404620309525
submitted by: dujella
submitted at: 2026-09-25 16:33:29 UTC
last updated: 2026-09-25 16:33:29 UTC

Witness: 13 independent points log in to add more points →

(3280729318743406140525683305, 136370513532909637993573542450302805757747)
(2004420905155556863813989325, 267302079008248225017960350719027728841587)
(4167302806802540767185656479, 2901166605421247653833323707845538575513)
(1594719740810079241267534139, 305235916593368997675449375746375537201863)
(4159068832062780178143336129, 9854475564997651078774897662499918040107)
(-368460890602040778403875111, -460747668449992266016725220241972232130637)
(-300349767685302549201917854295/81, -444016091633685419507358580969391174683002437/729)
(262296782150445561938555032425/49, 27868436696218284877796990576042324733548341/343)
(439602668904339871098967269241/49, 202393404152642614469373652108396656784837509/343)
(6009592539382691305043373561, 170405404702691084827965143341646595941395)
(2536517566056527740222292575, 215216887714970925692769354472305841893337)
(33680437410443933381567796925425/8281, 26112538631039482546133388969804078396623576377/753571)
(47416392206571753268924573660825/26569, 1246742575128868590454528409506092945429297864889/4330747)

Commentary

found by Elkies - Klagsbrun (2020)

last edited by dujella at 2026-09-25 16:33:29 UTC · history