curve #3171

y² + xy + y = x³ − 69993120101005305009860092170583378x + 7639258120193164015870793100033493923516729761682248

a-invariants: [1, 0, 1, -69993120101005305009860092170583378, 7639258120193164015870793100033493923516729761682248]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 151612699544290128712841885098258092358105179020910
discriminant (Δ): -3265242291676261456562481299358146094480365768839346133831338575858547413551869182384800263849907739312500
Faltings height: 18.9907
naive height: 252.4534
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 91291, 9932891205642543660883
regulator: 25808414385.152130790263733315833667475493053930619773795606481
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:40 UTC
last updated: 2026-09-26 20:40:40 UTC

Witness: 12 independent points log in to add more points →

(171777816398125539, 26167670979268269042638245)
(169740137584324909, 25477845876425289117707135)
(149332975795951549, 22740998019085332334924775)
(235735302889811599, -65111340369741228518429275)
(140624626342500187, 24029021933798151088331855)
(29075000559302299, 75025241273474528659949525)
(147635813086825357, 22884193456036867524732515)
(160487006141243383, 23233334933109112425738323)
(457571992790515204, 267236203879055542974358715)
(-21296509469728001, 95499782305256485473234875)
(26669222671152124, 76102328670260618345498000)
(161956158824337094, -23484430563596793664294915)

Commentary

Eroshkin (2006). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 14). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:41 UTC · history