curve #6111

y² = x³ − 2223553878809154769297149219646921217139027x + 1265762057777299845958349155768919629311019048671887761736582946

a-invariants: [0, 0, 0, -2223553878809154769297149219646921217139027, 1265762057777299845958349155768919629311019048671887761736582946]
c-invariants: [106730586182839428926263162543052218422673296, -1093618417919587066908013670584346559724720458052511026140407665344]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1186882284700177882316018479268883158412782043090562882320
discriminant (Δ): 11464976536649803948573156066163467288936659656709853563142454478331178530805912010448621964490930420726396510696519398544000000
Faltings height: 23.2012
naive height: 304.1366
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 37, 43, 59, 61, 67, 79, 137, 149, 157, 229, 379, 727, 1223, 4007, 6353, 9029, 10067, 19753
regulator: 764924589652663.2622881500575513146030127023900958383977866674320
submitted by: The Wren
submitted at: 2026-10-29 02:38:10 UTC
last updated: 2026-10-29 02:38:10 UTC

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

(-1706958221957946655203, 9364859099692985257788047428800)
(68227195200517007422, 33382222925420533273583432940800)
(930415507505245499947, 1539040687934489748718610808050)
(-3465680755762116227442227/23409, 142891552746843916011395000421155336000/3581577)
(1539713008537589717745/4, 172860744164781963798593744381073/8)
(30664267002866482432737/121, 35678164277011709384403424377434100/1331)
(35102254537098355925937/121, 33806372883159695077252116753153300/1331)
(440292612963698445178107/841, 381864644657063267347071918001978470/24389)
(981593525476561157858577/841, 393656654717950097160017997283718100/24389)
(335499389949578668181610297/516961, 3642808777051665101594471934635339260380/371694959)
(235081196816485636977027767517/253414561, 4702491568603569724206279284964161060989200/4034106396559)
(1851693642215855090972783800411/1819790281, 584798621679459229103126173894272205652305190/77630433597179)
(3779563399716471421697488566873/3081693169, 3350173565432006735513341804692279315291301860/171074032890697)

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=35816/160175 in the rank-five X400a family posted by wgxli with curve#4252: y^2=x(x+A)(x+B), A=q(121p-99q)(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 02:38:10 UTC · history