curve #6119

y² + xy = x³ − x² − 26162910326653698819009495144429969x + 1619921889148893606805302105402947471920510478441225

a-invariants: [1, -1, 0, -26162910326653698819009495144429969, 1619921889148893606805302105402947471920510478441225]
c-invariants: [1255819695679377543312455766932638521, -1399612512224644070628592388510947670833270102176345069]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 6372814782902779568050252265518228012820488033770
discriminant (Δ): 12509747942419163659993416753354658681701179718184007262176745415856042279249361874377847277423683522500
Faltings height: 18.6198
naive height: 249.3626
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 31, 37, 41, 43, 53, 71, 103, 173, 181, 223, 239, 313, 461, 677, 3361, 7001, 27647
regulator: 150689135619.65833158299943785914990258765921238144906856039253
submitted by: The Wren
submitted at: 2026-10-29 03:10:10 UTC
last updated: 2026-10-29 03:10:10 UTC

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

(-132949822306730204, 52424240101757686679982697)
(29058304948404049, 29735642939271256826818744)
(38902405976566175, 25709852605794879685936680)
(68486477919995191, 12220684249892278317449992)
(87686638217831566, 45918972655195128918817)
(99625484933114326, 1495270667617091119015597)
(114174340377956176, 11006381870339813300976397)
(130181060231320024, 20498756703226525400131219)
(-580372782000915733436/5929, 25998303245515766854657399578461/456533)
(-104656837510524386231/1849, 4295918448674154584576048370709/79507)
(-1524415693989731354/9, 932177689056560691383119159/27)
(24842619046314373129/64, 114429894231732613938747064169/512)

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=77011/124357 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_5 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 03:10:10 UTC · history