curve #6129

y² + xy + y = x³ − x² − 337872973491959196267x + 749512428602757726556083220155

a-invariants: [1, -1, 1, -337872973491959196267, 749512428602757726556083220155]
c-invariants: [16217902727614041420801, -647578738239802113470192715820545]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 55035751703278323680294108487782527741865489894062
discriminant (Δ): 2225856791490090803129792984855460575238958953485740217386811392
Faltings height: 10.8471
naive height: 153.4212
primes of bad reduction: 2, 7, 13, 23, 54611491665678687743, 240747418844039285759013869
regulator: 5281701513166674130.7204750743036580891528468651164509782738752883178188784
submitted by: Michael Rubinstein
submitted at: 2026-10-29 12:55:48 UTC
last updated: 2026-10-29 12:55:48 UTC

Witness: 18 independent points log in to add more points →

(-18528264265, 805628575796110)
(-11524538773, 1764293455553450)
(17498884123, 442102402103274)
(49360784767, 10214609492812602)
(1593154542315/4, 2008750985085040857/8)
(784894354887, 695180808913910270)
(-727989512064899/52441, 19967712224021608989828/12008989)
(-79781481273733/5041, 522682751579736273862/357911)
(-73073409534545/17161, 3266324369304531834170/2248091)
(-22411305980861/7225, 816514298624802752418/614125)
(-1894663561273/121, 1974588034260886670/1331)
(-1530711723653/81, 452778609176253818/729)
(190158042067/9, 47098912690667198/27)
(1326143585431/9, 1515413573203548190/27)
(2530987583859/25, 3961115940750459138/125)
(10653478494043/5041, 75841378842677664422/357911)
(1522814754009843/85849, 14613314378632321309570/25153757)
(2644651451997527/1369, 135998109120634475802622/50653)

Commentary

Mestre ten-root construction, roots 0,13,53,66,86,89,96,107,113,135: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 12:55:48 UTC · history