curve #5863

y² + xy = x³ − 94784358615791934862714682717293395x + 11227558448373333675717490400385599303475783503877025

a-invariants: [1, 0, 0, -94784358615791934862714682717293395, 11227558448373333675717490400385599303475783503877025]
c-invariants: [4549649213558012873410304770430082961, -9700610499394560302644385526270177108318534102994874041]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 55148200243517392566407009102465537396292625010
discriminant (Δ): 42098466258700710262503271270882020257211004135959985864481610114136483484857981490292131954624000000000
Faltings height: 18.8419
naive height: 253.2243
primes of bad reduction: 2, 5, 13, 19, 29, 41, 107, 173, 383, 587, 823, 1553, 2161, 5233, 67399, 1295839, 3574411
regulator: 715849892610414.58104536736162492758989838911477986168970746369
submitted by: Maksym Voznyy
submitted at: 2026-10-25 15:48:14 UTC
last updated: 2026-10-25 15:48:14 UTC

Witness: 12 independent points log in to add more points →

(6132556362321162514064978748585/300444304384, 15184944278517949334999296892074807550554183685/164681935673393152)
(27739341339538449593746422968303032750/153586825821789131209, -421744573394170402458654570803138226752301461549507965/1903403867389945671733298770427)
(521570308694111811659278910/845820889, 10644808352781598790162512964640097460235/24599008914787)
(35255902483027223116739983766/149765130025, 2566021724642668588722252582294664829458651/57958356494024875)
(1657028294770063076696129185774462120091206047041728600510/5972344293254415773538330163789439097529, -36597485125473910618893709193737629552513702854561227304016364712250432484474669643365/461548404401335819716453702448145221688958953807746492483667)
(31399825078319062580834421140/12230811649, -5525753272932802648785869327999245110979815/1352642152697857)
(8705178964447362091942142538240/43296281560081, -4918841485909364058025133238034398533076828745/284889142998798939271)
(111311673818677941045613816796008841425053734/95556234635129767780720609, -1136944400534899366423173909201726987648295679925076464861715854369/934089611857290925555131538235628431023)
(367858741277573760095430994510/1875193368129, 34725667135184574681771355671751553536059865/2567846668868385633)
(11852453075028085233186460/65561409, 411237019557667620437152792821321065/530850728673)
(88325561041162244961309467374510/486920668905841, -17985505584103626804591913629432128916377917505/10744527330942912235639)
(86059550195608561933099310/86359849, -764050884014916513445310614483500106915/802542076757)

Commentary

Voznyy (2026), T = -1411/2470 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-10-25 15:48:14 UTC · history