curve #6092

y² + xy = x³ − 1674225282754753659849494726931709279539x + 18041036144875943736394289523252772135860103679857786995984

a-invariants: [1, 0, 0, -1674225282754753659849494726931709279539, 18041036144875943736394289523252772135860103679857786995984]
c-invariants: [80362813572228175672775746892722045417873, -15587455229172815388365210368448737388892293199736211032656985]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 39859666157079499691041689625437974568946385526573501
discriminant (Δ): 159738946706579269556920638454117054758678150066292594269751017174036560949153635822058734073395179557481939933625651289
Faltings height: 21.5777
naive height: 282.5621
primes of bad reduction: 3, 7, 11, 13, 19, 23, 29, 53, 59, 61, 107, 113, 157, 191, 337, 373, 541, 547, 877, 953, 1153, 2113, 7723, 25889
regulator: 27819315179316.51292966624380944017979508662865020472755917302668
submitted by: The Wren
submitted at: 2026-10-29 02:11:23 UTC
last updated: 2026-10-29 02:11:23 UTC

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

(-44934311048500854945, 50444760890894982170526528543)
(-41343846407295598416, 128803527410077979391135510432)
(-1611847553655204115, 143998070843880432776030591282)
(259676734365180838707/4, 3421846263819753622727287336923/8)
(74957647499829361017, 560094414002426793485569082634)
(99236943747094801167, 910592433761113113189518216079)
(188033269603168189665, 2520204604506345671449125427962)
(-30519944076497489860739715/1697809, 455143618335686975336737920176005364199/2212245127)
(-47120265944688624409728/2209, 21789995731097739293318984060445108/103823)
(921940693413438966453/16, 21502952632620030452646873508029/64)
(119331197357438555238333/10201, 7761733799385233794261927727950266/1030301)
(54855113701106027350120689/5248681, 493586429371359637107736453711019539662/12024728171)
(67772720954835365230994068236/1539228289, 10404474541384341971025670955655687978603654/60388543462337)

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=154352/100399 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 02:11:23 UTC · history