curve #6116

y² + xy = x³ − x² − 18095532433380628256413630329343017x + 935726982905840470294972881222500303469371200539641

a-invariants: [1, -1, 0, -18095532433380628256413630329343017, 935726982905840470294972881222500303469371200539641]
c-invariants: [868585556802270156307854255808464825, -808468113230646162426221563766024558812192566128158125]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 9378724549861212877270871775515267856122067755550
discriminant (Δ): 969764227858417746459356351832868867356883786256940460036142668182770152846818800667458785477722562500
Faltings height: 18.4706
naive height: 248.2565
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 37, 73, 83, 137, 157, 173, 373, 479, 1613, 1621, 2467, 3359, 3847, 4831, 20407
regulator: 8841041346034.3120044111470963546407136148698836169960707973442
submitted by: The Wren
submitted at: 2026-10-29 02:57:31 UTC
last updated: 2026-10-29 02:57:31 UTC

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

(-111137283635712376, 39674974660427853156747963)
(12906044523379544, 26539309398050186993517403)
(40423110387746924, 16440856591342763440365163)
(335442975142049431/4, 22603762122076202978832419/8)
(181846875212790214249, 2452215568032358488616170483463)
(1362698571406802029/16, 224845945117859187341670217/64)
(8323633725744065788/121, 5398774054865233913081519729/1331)
(121713247191063036934/1681, 155666738309308046549235271253/68921)
(496719646480440116551/6724, 798542808712485994795349775559/551368)
(9356379974545171435927/145924, 348441744617075130638676345362749/55742968)
(4046624022300272078497/54289, 12949730349295875138410165099216/12649337)
(5560275355385172989321/100489, 324654797565460581155251628346384/31855013)

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=55264/65623 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_7 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 02:57:31 UTC · history