curve #735

y² + xy = x³ − 122776775069585351616747492319345x + 526395350922667203381059708825820468144096530537

a-invariants: [1, 0, 0, -122776775069585351616747492319345, 526395350922667203381059708825820468144096530537]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 1815436548573871882271810231932931339840134069673695222694630595510
naive height: 233.2879
Faltings height: 17.2785
discriminant (Δ): -1255524860496404041339919526715401858321042056672825094118416026325509307285490145973711360000000
primes of bad reduction: 2, 3, 5, 11, 17, 19, 29, 41, 59, 97, 2859137, 875435282615586462965097331992891038830823172451
regulator: 1109587422089427432.32640199927404202908593401462523859195613702757570250863210
submitted by: Valery Asiryan
submitted at: 2026-09-16 17:20:30 UTC
last updated: 2026-09-16 17:20:30 UTC

Witness: 20 independent points log in to add more points →

(-12764350695458866, -117825097573695156892237)
(-110539812835998374/9, -11504787835279316233210439/27)
(-3532900334322407464/289, -2199619652760180395337136301/4913)
(-10982967466937926, -741634620259801394859877)
(-10778781361429486, -772966755699875726184757)
(-9620216709360376, -903989618415689907437227)
(9362541474073688039/5476, -229757652001889507807081047943/405224)
(3876421835185682, -329712602243806909828405)
(304166893386438686/49, -20154148726831806625230151/343)
(56131302559111666/9, -1541155151533508181820319/27)
(6402704308585994, -52615685884698328523077)
(7832488525188362, -212730367850396509747525)
(9572262199783604, -477737440907691060150517)
(19384270952327228, -2330256673266826860485029)
(37296229629855764, -6915682639370567170263007)
(55601184018723752, -12868195635781433096407435)
(108481624363224362, -35550634907793508995211525)
(6209906915228474, -58611421435423497470677)
(6398500317489014, -52610637590405654589757)
(6307588257913274, -54052096076350389606997)

Commentary

Specialization T=-12/31 of the family identified by wgxli in the commentary to curve #302. Starting from 17 reconstructed generic sections, minimized quartic searches produced a set of 20 points certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 17:20:30 UTC · history