curve #2724

y² + xy = x³ − 14577831095019515866151496164805829803545289890x + 3972641278866872144327625339370539094362188991726331233427897011245892

a-invariants: [1, 0, 0, -14577831095019515866151496164805829803545289890, 3972641278866872144327625339370539094362188991726331233427897011245892]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 8802038786118636750817053683370
discriminant (Δ): -6619500831203115609617277340713252154956149208246653933036305165820042962327143869067935597280022464547570834532882118785504787824640000000000
Faltings height: 25.8939
naive height: 334.0388
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 163, 239, 373, 881, 160591, 146251529
regulator: 11876721.82441189459042287089219066343983826608
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:28 UTC
last updated: 2026-09-26 20:19:28 UTC

Witness: 4 independent points log in to add more points →

(16399897335548784918633584881066404096105451663369/2256331984482434191675456, 66405494185231859547269394364979695712749254309319171310698566700339237073/3389256983895918481537893117007248896)
(416309849925662980820155460688674632766256042912769261879004403672538136164/23228548709235437608973721548070279585371821836961, 8494059997256953732670839956975876985088584271961473372173611947955299331732705346963651193996863595696223759442/111952324321780282914319255060454964476975322608017115301511362848265314959)
(-114296281541622663062818562867160716/2667476164081, -292855287907170618614227992270993742380132657788434882/4356631437699816521)
(-208165126777807650411964955941779356/2434445593441, -257450455653443029799839564787424404943664385008238702/3798394860523782511)

Commentary

2-isogenous image of D-z10-021, rational kernel x=1509973410712173067946207/4. Source: https://dujella.github.io/tors/z10.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 20:19:29 UTC · history