curve #3184

y² = x³ − 9850452550387979749326269202x + 375740231494516934320326188325071932846796

a-invariants: [0, 0, 0, -9850452550387979749326269202, 375740231494516934320326188325071932846796]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 2283940066993910894016063059625994348127085137280
discriminant (Δ): 181342925315636466845645352743120757502743207149636907713607069167755482980139360000
Faltings height: 14.8702
naive height: 204.9855
primes of bad reduction: 2, 3, 5, 7, 13, 23, 47, 53, 59, 97, 107, 139, 181, 191, 197, 727, 757, 761, 809, 839
regulator: 4753988404611.14882092927599706315018965991346389283976023003482126
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:29 UTC
last updated: 2026-09-26 20:41:29 UTC

Witness: 14 independent points log in to add more points →

(59597179662347, -18966381943719671125)
(468341603149027, -9924223759723946107305)
(1490506586694163/9, 48948784237354600817855/27)
(3927182152411313/64, -24783741853059930038935/512)
(99675418637947, -619827702367609645875)
(404209324877147, -7901639581699037769725)
(16588293396294427/121, -1684606892467524851016765/1331)
(49698419359167682/841, -6322754171901949907580/24389)
(-111608524232053, 291354228825450335125)
(99482034251895654833210/1681246009, -484696087328740972706844703318208/68936130107027)
(1017897015741497/16, -5180987955984980058605/64)
(-14098873308549741854/143641, 34300367419186338970177363140/54439939)
(116165627047871378/5329, 161041286870798098866100220/389017)
(128707878782441962/1521, -22830246417322869214639600/59319)

Commentary

2-isogenous image of D-z2old141516171819-053, rational kernel x=-118188654139412. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:29 UTC · history