curve #6118

y² + xy + y = x³ + x² − 127295016846550791996723763268229844x + 17450277516693173309148420274473622685359019714251076

a-invariants: [1, 1, 1, -127295016846550791996723763268229844, 17450277516693173309148420274473622685359019714251076]
c-invariants: [6110160808634438015842740636875032513, -15077039774422901784930441181903495118970747809675673665]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2517462282771642323766182698181384519891875671381
discriminant (Δ): 462970536193400224696922612913682692054850013767544734805815794000063526507277861420766606438953569667649
Faltings height: 18.9702
naive height: 254.1090
primes of bad reduction: 3, 7, 11, 29, 37, 43, 53, 73, 101, 103, 109, 257, 293, 383, 509, 797, 1567, 2459, 8821, 47521
regulator: 69662985163567.170662227224802485374066342833615007321805893570
submitted by: The Wren
submitted at: 2026-10-29 03:06:20 UTC
last updated: 2026-10-29 03:06:20 UTC

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

(-63299254815556292, 158916110119092803436992991)
(172246695613895139, 25189395381022336205935690)
(220383699928661639, 10017044195410449267485520)
(243556476413036807, 29907570735512959880996472)
(-1842783552974198739/16, 11192420285362737063642920851/64)
(548208354408854105/64, 65489039191130580798003928599/512)
(114238777978325214898/961, 1883664259056947035244389521736/29791)
(3337445805128832959348/17161, 15845008863041831626370050165836/2248091)
(7144727198984075070592/32041, 70889294470652480414670294517308/5735339)
(3492419230219663608011027/24433249, 5633076901696138009843281362716587330/120773549807)
(1003195428303300658151808386/4641560641, 1842525532153999144035015348282367728741/316224884910689)
(294765203207024731492186656073/1122362261056, 54922048055516246217803964378445499466539361/1189048537158903296)

Commentary

Source parameter/search by bufferhe4d (2026), public Yukon submission 15009775-297e-4a19-9179-4a455b778310: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/6837b3bcd53a73fb5af3e271d91dec3148236631/submission/search/x400a-opus/xlog.txt . The source log records rank12 at T=2929/57979 in the rank-five X400a family posted by wgxli with curve#4252: y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q). The published section code credits 0xArion submission7e9f6673-cf5f-4cb9-82b4-a22add86d720. Using its four selected sections, a local seeded PARI/GP replay recovered the twelve displayed points and bounds[12,12]. Every coordinate was checked exactly on the minimal model; independence modulo torsion was separately certified by an exact rank-12 F_5 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 03:06:20 UTC · history