curve #3226

y² + xy + y = x³ − 3432055965319061202811800376695743x + 61269756155614728518291684114878312854267178414658

a-invariants: [1, 0, 1, -3432055965319061202811800376695743, 61269756155614728518291684114878312854267178414658]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 53142274232491098392872393941753394804628410908321062888347710
discriminant (Δ): 965557107383163947330266571531075387575273342835948360762619236684572481676928760057195262520556250000
Faltings height: 18.2831
naive height: 243.2690
primes of bad reduction: 2, 3, 5, 29, 31, 41, 59, 89, 107, 113, 191, 199, 397, 577, 1009, 1123, 1193, 1433, 1523, 1873, 3881, 58549
regulator: 4583088397164293.410590750293736906189906919849130157258410015280381031
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:32 UTC
last updated: 2026-09-26 20:43:32 UTC

Witness: 16 independent points log in to add more points →

(52359749829117744, -5011456011734531643879410)
(7195374267770659659/361, 6062270165442648153955195660/6859)
(236115388568601561399/3721, 2258329803580867130530672993840/226981)
(50703358678534041/64, -3917761687638900073132536745/512)
(-11476820464091303908997671824/2146743641329, -28038767068883385550809975218807601260699255/3145359408171500233)
(2368063282069676256/49, 986945335955161282760861170/343)
(-2447387879479090328559/43264, -77618369283490231354575667023695/8998912)
(3050112828892045803/3721, 1735433288646828013638157161268/226981)
(1855894021744395939, -2527058887223660374495154300)
(1433203571646947868107861195468691/5925086428023889, -52763291453960432414169609769428496000583489655320/456081057036098369813287)
(60056707550722616141348676651437904/255413572691007241, -14289421860151373856301539991843333049216594825264165/129082080905330591578736539)
(3259434083547439700396308311/6652522969, -184799626035881204338914001530255462840520/542599730920547)
(-53079669040417958915486886/824206681, -92239542195603261459597999793285725115/23662149604829)
(30722454113649421246579138284424618640163931/92361271221104324356723689, -167770006563695324352893199571176119972803728867667829490301785940/887635887434343098914013506069275300363)
(2271291652909427597718767275667199/30189726576151921, -79361401802522867086726690423434051554227883439160/5245522685173883300071031)
(1335365901612724479084981/23697424, -789236034633041293789844329354668145/115359060032)

Commentary

2-isogenous image of D-z2old141516171819-022, rational kernel x=528240160300186247/4. 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 20:43:32 UTC · history