curve #193

y² + xy = x³ − 87325226813408696187425x + 8446101555990886693844233482455625

a-invariants: [1, 0, 0, -87325226813408696187425, 8446101555990886693844233482455625]
rank (lower bound): ≥ 15
conductor (N): 2282558436678897130581398498347303110857569170
naive height: 170.0854
Faltings height: 12.1679
discriminant (Δ): 11801171435734791180357853315012481673861744917479863238791703372800000
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 31, 41, 675304453, 3187920097, 48610151546889853
regulator: 14786943101801.821816404552640176100785119072680650943651152261285874
submitted by: Seewoo Lee
last updated: 2026-07-01 22:42:39

Witness: 15 independent points

(748436483759858950/1292769, 572940742521140768169429475/1469878353)
(2457354167530/9, 1901167349508952825/27)
(1902067649424250/19321, 76122157752942148735825/2685619)
(101254141450, 25340801083372075)
(554406704154395050/5193841, 218571678161936581007716925/11836763639)
(90352672879870/841, 425494653096663013775/24389)
(109936667050, 13212042066356875)
(606701606153755/5476, 4639386552942000078275/405224)
(112298274550, 7470676593563575)
(144223305244433530/1274641, 5382887165331282577854355/1439069689)
(13557424351450/121, -11006174098335477775/1331)
(77564540933783050/734449, -12614392494911029526074525/629422793)
(1126143904425925/4624, 12671172546060460859525/314432)
(347065469002, 141223434230660971)
(1471166114339530, -56427728311403536554005)

Commentary

Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1265/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 15 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:09:50 · history