curve #2737

y² + xy = x³ − 115172563148557486438825854736557327124800x + 14946100635645484062544931966478780256593100221424795261440000

a-invariants: [1, 0, 0, -115172563148557486438825854736557327124800, 14946100635645484062544931966478780256593100221424795261440000]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 198635898246511938552934230
discriminant (Δ): 1272110915731690745347242481638837401450973515569524124338954055968158220987309997804049075262048906188654859903408537600000
Faltings height: 22.4515
naive height: 295.2554
primes of bad reduction: 2, 3, 5, 7, 13, 19, 29, 43, 73, 89, 251, 401, 23059, 203659
regulator: 8334825.648313526817406328337738842055002508491
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:03 UTC
last updated: 2026-09-26 20:20:03 UTC

Witness: 4 independent points log in to add more points →

(140342086040428584675412608/418609, 1013966543470775203396991234663843724096/270840023)
(4425926650616177303086321815900800/18803159460121, -76392275099587711515891159367569111208042997032000/81535407043703747581)
(-2152149259851267298975951235495723247237484928/27685083517982056550547801, -705098987456847947941186490177264401080807904054673294914019668982848/145669542746889035913527501875435636701)
(1838798187659289542681484337122555682877874592180248898701962961671848337086263936/2615436572771090054258094004448468304141400719211811235996481, -70964978023335271703983015484762014467750085211958053012866787750470416177802948640232348206573656878998259106512863224128/4229765480357025640060547676956946311445560394508305311739193695401794624473430631510289121)

Commentary

2-isogenous image of D-z10-012, rational kernel x=3132697862812686052351/4. Source: https://dujella.github.io/tors/z10.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 20:20:03 UTC · history