curve #4172

y² + xy + y = x³ − x² − 2507090808474296078x + 4078438750025564818217988381

a-invariants: [1, -1, 1, -2507090808474296078, 4078438750025564818217988381]
c-invariants: [120340358806766211729, -3523771079480556388309894008633]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 30066338547903581129348879285312931560456912310
discriminant (Δ): -6177209130104430645594498722574206322842373406341019238400
Faltings height: 9.7749
naive height: 140.6742
primes of bad reduction: 2, 3, 5, 7, 11, 47, 499, 6709, 80248601, 467823647, 3179504860341203
regulator: 154622977862451.8101235780512184581624797136829236358509839124554133113
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:57:48 UTC
last updated: 2026-10-11 22:57:48 UTC

Witness: 16 independent points log in to add more points →

(1663310025, 67156322735027)
(1230965598921, -1365742144156772989)
(281390890659609/172225, 4710557310624278314893/71473375)
(86615885504745/8281, -798279623229000297783/753571)
(-1035163129351/9409, 60215260052280484563/912673)
(296873776691105/160801, 4886355868345728507987/64481201)
(-223238782875895/165649, -4771752286378817056539/67419143)
(9428328112625/5929, 29277916161396534591/456533)
(-974396079136295/690561, -39785179924648266108883/573856191)
(4588794004859/6724, 28571728716609191853/551368)
(497203104619431629/444577225, 484671279127058106342951147/9373910789125)
(9803515920875/2809, 28959483192488153229/148877)
(3883327577, 230006903413011)
(1649003448811321/1640961, 106643285105852243468371/2102071041)
(363366677, 56704515982431)
(1353534905, 56255492437587)

Commentary

Mestre construction: sextuple [-494, -490, -156, 308, 342, 490], T = 413; y^2 = r(x) with p(x-T)p(x+T) = g(x)^2 - r(x), p(x)=prod(x-a_i) on the locus sum a_i = 0, 12 p5 = 5 p2 p3. Found by Nagao-sum sieve + ratpoints search on the quartic.

last edited by Andrii Prokhorenkov at 2026-10-11 22:57:48 UTC · history