curve #2731

y² + xy = x³ − 68103683581864090989576867388461910574953624420x + 6802506970429183880681255237338026167411750400145087841636060266672400

a-invariants: [1, 0, 0, -68103683581864090989576867388461910574953624420, 6802506970429183880681255237338026167411750400145087841636060266672400]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 4944555496787281270125087159390
discriminant (Δ): 225427872764028158027822043954604207698285139456368844840188670142237027714872895654341764709837516521231480517478382035564545021221705600000
Faltings height: 25.7676
naive height: 335.1257
primes of bad reduction: 2, 3, 5, 11, 47, 71, 103, 131, 139, 223, 239, 271, 1669, 99312839
regulator: 15391455.55919110033359406684474687083565079395
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:47 UTC
last updated: 2026-09-26 20:19:47 UTC

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

(245075709834578877903979679225/107584, 120561400968894713290183540866299017304659985/35287552)
(20473310573592798046803827036852632058/3189842436169, 92561080744653782349478699345030612309595839324667644264/5697100058949504197)
(102839472237203803196117557631087681183329993685901200900/528402951416503165496948590712689, 368403685868726986172249667520602770328327557954814190817893525580394841821477901840/12146407636925224163807205149831900881679557119863)
(5038493256669721878718449639554813596409752130/45632446952519585956561, 244469898845881262256421267529153447721897991352746938670742781498010/9747890373994798262911231869043559)

Commentary

2-isogenous image of D-z10-019, rational kernel x=-70687215479339285336760. 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 20:19:47 UTC · history