curve #6115

y² = x³ + x² − 2072924090571698036171537138938529x + 35964524180423414712681640385107390364794707794271

a-invariants: [0, 1, 0, -2072924090571698036171537138938529, 35964524180423414712681640385107390364794707794271]
c-invariants: [99500356347441505736233782669049408, -31073348891885830908759075377381819692585323548546560]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 977582092190237829070067009190555061574831808
discriminant (Δ): 11303499691733521737850035966749094517714554982954090728765321160814027894475538035382762336255279104
Faltings height: 18.0114
naive height: 241.7564
primes of bad reduction: 2, 3, 11, 13, 19, 53, 59, 83, 101, 107, 113, 127, 131, 263, 541, 653, 1949, 3221, 4721, 129061
regulator: 39598077723446.514568519932382123032217947776972603561041753247
submitted by: The Wren
submitted at: 2026-10-29 02:53:49 UTC
last updated: 2026-10-29 02:53:49 UTC

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

(-46999446370778027, 5437966316885264379585060)
(10380718040479474, 3945212625380131467544875)
(113020952208516277, 35289948068652460905115380)
(451006563891795775, 301394954366595962249341536)
(-3790030437584244160003481/87254281, 5409557508130450140149818167246701280/815042238821)
(-38220317764658385047/3136, 1353704119935528125442364424235/175616)
(835619932553601/16, 383232637433418846653711703/64)
(1309781876686790521/25, 1054509321698772586705704744/125)
(4137656608551344575/49, 7380732736162855067926388448/343)
(210136272591919518415/5329, 1533469411088830786273426937472/389017)
(137876030608667166668863/2362369, 38731410202568556912025946202946080/3630961153)
(4695054606190088966523081/301473769, 14295720872454252294474874042754486520/5234489051147)

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=51931/81150 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_7 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 02:53:50 UTC · history