curve #6096

y² = x³ + x² − 133501174791446756167798462754401075940065x + 15332595731354112539105014837463510696561154827086096773221663

a-invariants: [0, 1, 0, -133501174791446756167798462754401075940065, 15332595731354112539105014837463510696561154827086096773221663]
c-invariants: [6408056389989444296054326212211251645123136, -13247362711889953233825181157908409907605163727875655121934255616]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 172693048172172746400478624258257772692731498377790306880
discriminant (Δ): 50719095537321448486800593157100404934230977199779466030406948057247732050876048362567274754770441269937704153912075264000000
Faltings height: 22.6458
naive height: 295.6984
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 59, 73, 131, 211, 331, 479, 659, 2447, 3593, 3769, 4721, 24469, 57737
regulator: 445317368017876.8655270644879605820556204719686065986455254220463
submitted by: The Wren
submitted at: 2026-10-29 02:22:48 UTC
last updated: 2026-10-29 02:22:48 UTC

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

(-330283774209081176209, 4836949330242889465120675611360)
(128903579514384143471, 515456524321435476243858794400)
(331570117565544088961, 2742261367287858543362131795080)
(511309597216888250783, 8985970665084869074186730049888)
(572414222952040519871, 11245921410467874953108759054400)
(736970563116866485871, 17810495419198384855528263626400)
(-577984271452349146883593129/1819801, 12443979105891650904095682740785162744800/2454911549)
(-173863889526224766295340569/1413721, 9189988924108047777441261100034611873920/1680914269)
(-31615638174476359771507969/192721, 484676964432577689134706955960181838720/84604519)
(4073048705057645142562/9, 186289891113480928064364431532889/27)
(165120132157853634795161/256, 57571904728770442402488373808899005/4096)
(516277535192655563386877450599/1287159129, 236888155140465681055434722864336622478106560/46179408071133)
(59828651045744227502539841125151/455367186481, 75565020487319621516801951621428649325034651200/307285875742057129)

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=11017/30600 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_7 reduction matrix at thirteen good primes.

last edited by The Wren at 2026-10-29 02:22:48 UTC · history