curve #3614

y² + xy + y = x³ − x² − 3565267067359985442067065858437x + 2601059096758546793470762667503415178211022461

a-invariants: [1, -1, 1, -3565267067359985442067065858437, 2601059096758546793470762667503415178211022461]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 4676874174796470773416020830070114756053796903425726941833510
discriminant (Δ): -22311124624690959275070379946838892108196808696456112997145794406036039598955529515520000000
Faltings height: 16.3805
naive height: 222.6676
primes of bad reduction: 2, 3, 5, 7, 11, 19, 29, 31, 67, 73, 2657, 6869, 12823, 53731, 590041, 2195876051, 495817494960962003
regulator: 2398678809590617282348.939480971291982573348394892842241946097055059402571858
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:19:35 UTC
last updated: 2026-09-29 19:19:35 UTC

Witness: 19 independent points log in to add more points →

(-4035468161546801/4, 575282993300551131437047/8)
(1054975905594751, 3734857902826730923904)
(1421894538852991, 20159093159524809503744)
(1517098651692379, 26152126715118708732704)
(49438990988456671, 10984798715620381964013824)
(-6799943126875076561/18769, 159448526204414460536231469952/2571353)
(-78306058195767689/81, 52286351306067454884040736/729)
(7700715620100055/9, 359122041911391998068064/27)
(11152281158257711/81, 33508588206793388230094336/729)
(52782719537125519/49, 1111085285404896192490112/343)
(117908245458862439/169, 46772906677458739631576048/2197)
(185854137748312439/169, 7033144099287252551810048/2197)
(381079020296745569/49, 228876946242866716277266412/343)
(575819901757041431/361, 213892886492118012344511616/6859)
(36762809583368746759/6889, 210511650962795611718236460608/571787)
(153517881489488677999/120409, 472664743874217314242686701152/41781923)
(708573019102153441639/582169, 3573631971364685200974967547648/444194947)
(9404609085806083568839/635209, 904955252883921174311806691994352/506261573)
(86282090478964775537639/60637369, 9548798453897790651965939030820352/472183192403)

Commentary

Specialization at T = -1/11 of the rank-17 fibration of X1092, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #3432 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:19:35 UTC · history