curve #6094

y² + xy + y = x³ − 19698994491882803589762665012791223529526x + 962856804971518299178057995326904693560129272196973182611448

a-invariants: [1, 0, 1, -19698994491882803589762665012791223529526, 962856804971518299178057995326904693560129272196973182611448]
c-invariants: [945551735610374572308607920613978729417225, -831908279495391810491260435565861217094414603059105797870417125]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 25297518055926931121987512801766887116408932913900450
discriminant (Δ): 88724678104095664145449528930961189523406811635351857989659359051289720704905717605738595479527388007257188560349506250000
Faltings height: 22.1345
naive height: 289.9578
primes of bad reduction: 2, 3, 5, 11, 19, 23, 37, 43, 79, 107, 109, 149, 223, 653, 719, 1567, 2083, 2857, 3719, 50383, 113539
regulator: 59907429792130.34519496728688669902332119842546759181126695131137
submitted by: The Wren
submitted at: 2026-10-29 02:19:09 UTC
last updated: 2026-10-29 02:19:09 UTC

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

(41074468147329062797, 472258730504714428034689156601)
(737920494842267247862, 19703906846106647496847650031556)
(-904998812287278735431053148/11978521, 58920890472294434788602243430795661261536/41457661181)
(-197142810043642732097552/2209, 147197635242562628681471438972438248/103823)
(-55967887156324823291987/529, 16605755523526031320876156055633767/12167)
(355393161216981872008/9, 13406654055894586281267172079512/27)
(25309975443224983352623/484, 2930263779062245407175172750196323/10648)
(55307792112593068842298/529, 2613859036338549966407277453084127/12167)
(240166285738040772388933/8464, 508682439139234671586405805474272363/778688)
(2355504027296051119791232/3481, 3542263700241025973748777246151979029/205379)
(30825628084124178491864332/36481, 168907402858021564736304261703141695421/6967871)
(34581993801022948499965888/277729, 97149217698693634830357598935083132233/146363183)
(47882163339110176627489533913/1145687104, 17879231159775024786711509609371274624747167/38779217096192)

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=151149/77071 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 02:19:09 UTC · history