curve #6083

y² + xy = x³ − 394181060217954067917187260588x + 92159762907372060623719986225470248351690167

a-invariants: [1, 0, 0, -394181060217954067917187260588, 92159762907372060623719986225470248351690167]
c-invariants: [18920690890461795260024988508225, -79626035151969488759930403791499184613343066625]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 4128396631990771588539528525089728239975 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
discriminant (Δ): 250671752190444715261009787462482561212827747286557989963140556549797013673299684766015625 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
Faltings height: 15.9232 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
naive height: 216.0534 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
primes of bad reduction: 3, 5, 7, 11, 43, 47, 59, 97, 107, 139, 257, 263, 463, 719, 953, 1409, 1831, 6829
regulator: 24927168014.084435347376963484702691000444373870624336143132338
submitted by: The Wren
submitted at: 2026-10-29 00:53:25 UTC
last updated: 2026-10-29 00:53:25 UTC

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

(202588408910106, 4540693174288439525055)
(139039144488582, -6327800168454628534791)
(-11464016697595563/16, 166376384670918425006751/64)
(2731780310056853137317/8994001, 18182288391606089787856419261621/26973008999)
(11008241374107, -9371331642216810656241)
(-533399027949297/4, 95449242362537883568797/8)
(1057058866472982, -29268015283907533630491)
(146195129738385704523/218089, -1157444798086815586738366437558/101847563)
(-729621108047975895603/8473921, -276300284624670099753085239192771/24667584031)
(1444422197726592618/5329, 889703324326134040692360843/389017)
(423650327359107, 1096092369624693207759)
(26383703446190215098/1369, -135449017300421513559398655513/50653)

Commentary

Original specialization/search by kaankolcu (2026), public Yukon elliptic-rank challenge submission 43af8780-369f-484f-b8df-47147017654b. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/f1c42abe0153e017898256f2d1ad9dbe7b47cd32/submission/board/2x2-12-own.ec . Specialization T=771/1409 of 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 twelve supplied points have been checked exactly on the minimal model and certified independent modulo torsion by an exact rank-12 F_5 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 00:53:25 UTC · history