curve #6090

y² + xy + y = x³ − 530251823344889840555906834632151x + 4662610669809163238655232938992868501361090224698

a-invariants: [1, 0, 1, -530251823344889840555906834632151, 4662610669809163238655232938992868501361090224698]
c-invariants: [25452087520554712346683528062343225, -4028495618715117076376252540121906905201274047654125]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 176767773759214183330795954037255316905199450
discriminant (Δ): 150062664677126979912499363523602717564908962224849039931331616964566267665117629271709953225000000
Faltings height: 17.6605
naive height: 237.6663
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 43, 59, 67, 229, 383, 421, 449, 2357, 2689, 3389, 3943, 6311, 26951
regulator: 282806393045.78914705804919953124316301454759889275608348149430
submitted by: The Wren
submitted at: 2026-10-29 02:00:36 UTC
last updated: 2026-10-29 02:00:36 UTC

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

(15141542066359071, 324385023498662502880036)
(63261472779400203/4, 3855930182870274949795043/8)
(15937627958664951, 509842269190522744699492)
(69299204608735607, 17341213262736004045496196)
(185666733913702482, 79413735519369962395954321)
(-4125566531201819913/361, 20838115571003502431823399244/6859)
(-2042773066518952964/81, 1029728796856227435624145793/729)
(139434016767277078/9, 11006872764154083643079432/27)
(917203735285276572/121, 1382415809108761495075550126/1331)
(53789318156955902862/5041, 167692778793674232420709888631/357911)
(99185178956583917967/18769, 3643762399526750559309063489092/2571353)
(125710893444788912285487/13845841, 39798558273260242621425689227705756/51520374361)

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=181632/180163 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 02:00:36 UTC · history