curve #3654

y² + xy = x³ − 74478614497001051213037130703918863x + 7813308578453199795281011676368530519633974132260217

a-invariants: [1, 0, 0, -74478614497001051213037130703918863, 7813308578453199795281011676368530519633974132260217]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 4037723001611237548976485813408801349432131894744731730350
discriminant (Δ): 68163435661531471885519549668862676346305753044573207755789547466526004476749194512786826762265664000000
Faltings height: 18.8246
naive height: 252.5011
primes of bad reduction: 2, 3, 5, 7, 11, 19, 41, 71, 79, 97, 151, 163, 179, 191, 193, 199, 239, 811, 1481, 2179, 21563, 392599, 438143597
regulator: 6102959932109307084.566269323407833189864264185422777930493446259300240
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:27:37 UTC
last updated: 2026-09-29 19:27:37 UTC

Witness: 16 independent points log in to add more points →

(-310720537090836082, 30921345983613377367432077)
(-237906260208257158, 109849389448240194789714629)
(-154270142490639358, 125026466140984196544965429)
(177288480792769598, 13472149756402112242668845)
(1159975142075985683/4, 823743855739228492917291667/8)
(586329223354907438, 407079511491991301643000965)
(-4268142720661780865443/913936, 78930722991572916201191293642622029/873722816)
(-363974277303453525292/5329, 43635200499636613673578116110933/389017)
(-7055503944221450914/361, 660086647903860703699062894095/6859)
(14264963340788166713/64, 24479482998833027941910176273/512)
(2764409561766147910958/11449, 76572041912665819822750773512047/1225043)
(22230144638597768803238/214369, 3445401120213003698384902728765203/99252847)
(27732438078959310770542/201601, 1183778864131379436468527939435021/90518849)
(151529577575708468807468/923521, 2841093182264491090118066520788327/887503681)
(336763059968161217203582/447561, 184033774648337232701656011140699521/299418309)
(275188664581017270069721448/1903315129, 681477300669137981971591410261652163327/83035929132883)

Commentary

Specialization at T = -115/1351 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (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 19:27:37 UTC · history