curve #3212

y² + xy = x³ − 1613802768862644293121468453997350x + 26289607011924700896029177487385187368561005199332

a-invariants: [1, 0, 0, -1613802768862644293121468453997350, 26289607011924700896029177487385187368561005199332]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 31049725562981000812793039146883386111013163212447670
discriminant (Δ): -29586932819498736480743492690842521357191417462665474052399181216753024498921118032854290329600000000
Faltings height: 18.0321
naive height: 241.1097
primes of bad reduction: 2, 3, 5, 31, 43, 53, 67, 83, 97, 113, 227, 271, 397, 449, 569, 881, 929, 2909, 4877, 29347
regulator: 15641852200862617.995471302700793414999128626716564384691874167012906
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:55 UTC
last updated: 2026-09-26 20:42:55 UTC

Witness: 15 independent points log in to add more points →

(15453402734939004, 2245273645004442407027898)
(1720848148022084796/49, 1233322841847477584225468214/343)
(432231907918688748, 282984502929354572201438346)
(519899984009836556/25, 164063922568029433066692706/125)
(697377761407753511361/16384, 12356265034494147623890694426721/2097152)
(2137580594840195484/121, 2415512461911337387368261438/1331)
(2022545054740654098396/413449, 1143829169598773980778724557350686/265847707)
(-130499690913801610164/2809, -148330119940849268221723230654/148877)
(-446556582555143887716/19321, -19224645113439615764187068710338/2685619)
(563579324167136685725480586396/21915440420449, -137469464210196932390476074726869599722689814/102594789376207005457)
(393671470259812850147335159724/5404504408081, -215878771248534826385650234037232468268403258/12564170263225977479)
(420573481298932770774732111084/24394817470321, -228340464638856226372266155110406379050969562/120488711310654624631)
(55936880181194701864655921943836174124/1745941936607226541681, -199414740524496347897328366200363175170962146637616629542/72953258543490178614969782278921)
(-4302865350045018513213664/179479609, -17205081039690217633856102263970666846/2404488321773)
(2450244716637695959762185060355625753766/9283565747381508863809, -119961553841626692118723150715856525998119669647620224409964/894483278433101995948780462692127)

Commentary

2-isogenous image of D-z2old141516171819-049, rational kernel x=373285797610436767/4. 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 20:42:56 UTC · history