curve #6093

y² + xy + y = x³ − 16019841549717559454010641583424932807174x + 762251487039088878735945234487630274725620019016125376426116

a-invariants: [1, 0, 1, -16019841549717559454010641583424932807174, 762251487039088878735945234487630274725620019016125376426116]
c-invariants: [768952394386442853792510796004396774744329, -658585284801772791229010111188892221643624462623938920394280933]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 114103645867500682535751079990457905333656368701852230
discriminant (Δ): 12116655417294506724487067006719893640138228080923445645031944877742036769444156386803484175768882679761907066289687000100
Faltings height: 22.0146
naive height: 289.3375
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 31, 37, 53, 73, 79, 101, 107, 163, 173, 181, 229, 271, 281, 367, 463, 509, 691, 1163, 8887
regulator: 161881321413399.4481702196355366231964734657070231866002914006708
submitted by: The Wren
submitted at: 2026-10-29 02:15:36 UTC
last updated: 2026-10-29 02:15:36 UTC

Witness: 13 independent points log in to add more points →

(-131252575157858578654, 777031228505537957839595170077)
(-67147583879024772892, 1239027944808352742309123588268)
(39549235010832527660, 436508451105217225093570363767)
(61892176611835632833, 88517604990529838294653891023)
(254875953349476204827/4, 110115120710285041065208177599/8)
(94926455997781744106, 311335896221864833733303170896)
(145396218575344237271, 1227481951998366728448872900427)
(-63503867479316626806276495524473/1038538313569, 1301909739439335864781459182584801580227717697429/1058360894360091503)
(-1658755185287313605112652654/12453841, 32089716048725894031013799626343909058144/43949604889)
(-289425569065675792910008/2209, 81221970351954089685905503952137884/103823)
(96674615104871941982417/1024, 9901423488723660823379865279079689/32768)
(3010590946052200914867278/11881, 4664482237066542807757715548483533912/1295029)
(833477272224325889494472603/14280841, 8715676621941224163393125179114883750877/53967298139)

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 rank13 at T=172128/135161 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 thirteen displayed points and bounds[13,13]. Every coordinate was checked exactly on the minimal model; independence modulo torsion was separately certified by an exact rank-13 F_5 reduction matrix at thirteen good primes.

last edited by The Wren at 2026-10-29 02:15:37 UTC · history