curve #2746

y² + xy = x³ + 998728507830029646107433718897840336790x + 1243163821107728868158012809479300726498983016724583277256100

a-invariants: [1, 0, 0, 998728507830029646107433718897840336790, 1243163821107728868158012809479300726498983016724583277256100]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 130572366625335476548016310
discriminant (Δ): -667700871783801874721052941269667033361457179551755063346361335409848824577857143348141911278626383591709019916910000000000
Faltings height: 22.2468
naive height: 290.2687
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 29, 31, 41, 71, 107, 139, 19139, 307201
regulator: 3531653.887524334874223740200316821836315134412
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:28 UTC
last updated: 2026-09-26 20:20:28 UTC

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

(1973363274330295729882070/17161, 3814165158023360504669789181087271840/2248091)
(207599997201532638069390943129971236/418177346878225, 95258918573915288903291132411895798505965025756539286/8551469564591371141625)
(43172187068554287878855271247034919323312902170230/3937426871643394829738235649, 283666522106199648130785059642478944336860202288258513748700987232930426320/247069280965865753800943971650286847435393)
(-5078207909635260004925865880/49126081, -64738254149283108365690160230273524079290/344324701729)

Commentary

2-isogenous image of D-z10-010, rational kernel x=52214884248122398068. 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 20:20:28 UTC · history