curve #2740

y² + xy = x³ − 18193823476120703293662493348661828678758235x + 29869921319173737793236316986037320276015494544271523285414982225

a-invariants: [1, 0, 0, -18193823476120703293662493348661828678758235, 29869921319173737793236316986037320276015494544271523285414982225]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 230951513872748392384304490
discriminant (Δ): 1462541505135558063664188468471620315305637335655123410877608134534916306534678069319441211520063572633037916160000000000
Faltings height: 23.1918
naive height: 310.4426
primes of bad reduction: 2, 3, 5, 7, 11, 19, 37, 97, 2129, 2311, 21751, 13700201
regulator: 8688163.004085452925959092779969703389949228367
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:10 UTC
last updated: 2026-09-26 20:20:10 UTC

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

(497674581787641293286798666328769897219054396745124583230/96571539575500036352466173858254369, 8106913835245590599446838664152576520439033941641528781546419435310998594705270520895/30010534654303954438568763386496765003786461928685903)
(811697797494060056628408862816/329604025, -312529898961549912894119367867471609301/5983961073875)
(36624881956695744067755428940511307856855735878330363944631073684765318430/14763997228363931516286208838045290292208423611997649, 2786027784200420762852982426695450129074720317189638409562001895239644459516060549607505456737761557282089495/1793931816791384351502278739211705153957846441312350276320925801239258202855143)
(117236815744058057404943512899840743064670/47918610527298187681, 457403475565314842072507593063676616049015020027690111200735/331708289490937450016027992079)

Commentary

2-isogenous image of D-z10-013, rational kernel x=-1231319839060116714082. 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 20:20:10 UTC · history