curve #3216

y² + xy = x³ − 2360001638573495470166776973829890x + 44062209861456503824552700986268012124494211867396

a-invariants: [1, 0, 0, -2360001638573495470166776973829890, 44062209861456503824552700986268012124494211867396]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 25369082817839617747015394004038159673868651735600634772174
discriminant (Δ): 2515494269398113961594848355908555284033734064445209719662471687323901833820855380255187307706584512
Faltings height: 17.9672
naive height: 242.1455
primes of bad reduction: 2, 3, 11, 13, 37, 41, 47, 79, 83, 101, 373, 433, 467, 733, 911, 1289, 1447, 1949, 2011, 29629, 31751
regulator: 171611684664390649.62769239138963022116704391418896957861408574480279
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:06 UTC
last updated: 2026-09-26 20:43:06 UTC

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

(138695773158732088, 48834041585953407884183434)
(-21427735520031344, -9208322426930304955308494)
(122024281723192576, 39661280154856350066892738)
(29242915451430202, 236442760796635738944484)
(27013575629484486, 151102416796596076503138)
(27147999645803296, 36590914487122413608866)
(27064561308150808, 119725915735084111653694)
(325408935966906376, 183668087126791093681170238)
(32346505156149730, 1252372313833603977888796)
(8017811252386926064/169, 13699439601885268151273553082/2197)
(25919578616563216, 552604042585419645044146)
(58444300755232240, 10284156882845797880482366)
(5297168817039049090/169, 2065916661763672627482293482/2197)
(42396309523743608085952/1212201, 2768805689186555149986676959183254/1334633301)
(27689547135677436230486824/2809, 145704773233526650421493285148485147026/148877)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 43). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:06 UTC · history