curve #6081

y² + xy + y = x³ − x² − 4282225273137684416x + 126036726550406944908943925706

a-invariants: [1, -1, 1, -4282225273137684416, 126036726550406944908943925706]
c-invariants: [205546813110608851953, -108895731738626639742329811976425]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 5126661320120532183815901
discriminant (Δ): -6857405174986517490508887687232491385700057838363709500206991
Faltings height: 10.3524
naive height: 147.5359
primes of bad reduction: 3, 7, 11, 37, 67, 73, 181, 397, 619, 5113, 179749
regulator: 21364557.331725510364748301236664310842245324729187550944
submitted by: The Wren
submitted at: 2026-10-29 00:44:26 UTC
last updated: 2026-10-29 00:44:26 UTC

Witness: 9 independent points log in to add more points →

(-3534417573/4, -2874776619045231/8)
(13486626584/49, 121210032838761210/343)
(-4330494862, -251732430197682)
(-3361414501/4, -2873788360621815/8)
(-2035564771/16, -22769920934179125/64)
(131965178679011/255025, 45343345104505522429194/128787625)
(46414863876, 9996008833864445)
(166790271282227/5476, 2153877828887456488257/405224)
(-264581223159/64, 138417048587655145/512)

Commentary

Original search by mpjunior92 (2026), public Yukon elliptic-rank challenge submission 5efdc131-d52e-4b2c-9962-c0b912aa9de8. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/07e2abcc754fd889214af458aeaeb016a2004705/submission/board/4-9.ec . The source used a Mestre-Nagao sieve in the Z/4 Kubert family y^2+x*y-t*y=x^3-t*x^2, with t=u/(D*s^2), (D,s,u)=(3,29711,-2980125361). The nine supplied points have been checked exactly on the minimal model and certified independent modulo torsion by an exact rank-9 F_5 reduction matrix at ten good primes.

last edited by The Wren at 2026-10-29 00:44:26 UTC · history