curve #3222

y² + xy = x³ − 12519641432143669965561465014249235x + 496609721939623321258971611666543569867328038152225

a-invariants: [1, 0, 0, -12519641432143669965561465014249235, 496609721939623321258971611666543569867328038152225]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 6098008034408866436674675605910737655832159836307310
discriminant (Δ): 19049804055740422636703369496928562404173465091538516065578054962756653138580094345142385291506483200000
Faltings height: 18.5580
naive height: 247.1514
primes of bad reduction: 2, 3, 5, 7, 17, 23, 29, 31, 53, 67, 97, 131, 191, 223, 227, 313, 433, 2609, 3881, 8689
regulator: 3090058055940796.3253886660477428457400076243443319492820062357139849
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:21 UTC
last updated: 2026-09-26 20:43:21 UTC

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

(3876599804806421550/49, 392442974506308727738336095/343)
(714050068010105970, 596345437865988815579260065)
(3248403062355795311070/34969, -76039752690198980701225646312085/6539203)
(80651201213571870, -3389795941246542410660535)
(78374809310721818110/9, 693791631302845413226223174915/27)
(87407024296498830, -8372199681770399293527495)
(39911201720738190, 7778849576579266716434745)
(24190783127733767538510/134689, 3142531100560448854102657964259735/49430863)
(159251310117558838198309230/1538443729, -1062146199374335409558136667265313123345/60342378382567)
(462003408252321052942841978430/5737074057961, -44829337602352787106906899941411898267739405/13741548788035288459)
(24593180254728234018000603086436030/1339423832347031209, 25609423577189935787902220257145516983002293532998445/1550161809106313397820679573)
(-371681472883039542422762826723090/6841781647093969, 18042201966096829006317416882718337813321163267055/565918399971621412755047)
(517391138415798245574787882263468030/3347926558329084169, -290746472473820319855726098992532914726025760915268995/6125815096800753751159140197)
(935098301013470434070417127870/10869543016201, -268790568963469755246935325459383283425031165/35835785500570060699)
(2575237184777831609055615390/19491789769, -92232810223904040199195781282788119619395/2721307245019397)

Commentary

2-isogenous image of D-z2old141516171819-050, rational kernel x=-393203196144086129/4. Source: https://dujella.github.io/tors/z2old141516171819.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 20:43:21 UTC · history