curve #2315

y² = x³ + x² − 16388459788431249768180x + 573404154184717410734999527857984

a-invariants: [0, 1, 0, -16388459788431249768180, 573404154184717410734999527857984]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 12853118643121903611159945035499392612472 ☆ record for ℤ/2ℤ torsion, rank ≥ 14
discriminant (Δ): 139666611043087056720755188897624419869125934752578076121913814870272 ☆ record for ℤ/2ℤ torsion, rank ≥ 14
Faltings height: 11.7823 ☆ record for ℤ/2ℤ torsion, rank ≥ 14
naive height: 165.0662 ☆ record for ℤ/2ℤ torsion, rank ≥ 14
primes of bad reduction: 2, 3, 7, 11, 17, 29, 59, 71, 73, 79, 109, 163, 13306129, 45371593, 54443929
regulator: 8833567633192772.94582791106285634093459235715363923942517567056843
submitted by: Steps Unbounded
submitted at: 2026-09-26 19:53:55 UTC
last updated: 2026-09-26 19:53:55 UTC

Witness: 14 independent points log in to add more points →

(319676299710, 167341153930441722)
(-19818388858, 29839788596099874)
(709956150918, 588882846967431390)
(-400925437359234/2809, 329699039874202092066/148877)
(-509365626379/4, 195204358330507239/8)
(-68649249161046/529, 276036935034835780722/12167)
(33808314006906/169, 159956248766856664374/2197)
(158736455818048/9, 1999878454176200578232/27)
(8767645419505146/1369, 820801536534807883376454/50653)
(8206459694805588/73441, 233267797729074510843840/19902511)
(26307174391240686/1274641, 22476958728235001021549574/1439069689)
(721271790628013274/6027025, 267178927406490005261475582/14796346375)
(10525125867356841/15376, 1061724665191682198659491/1906624)
(-22976111690685530977146/271752647401, 5214012095773271554235730572667738/141664383337493899)

Commentary

2-isogenous partner of curve#2177 (found by Alvaro Lozano-Robledo). The rational 2-isogeny with kernel generated by the 2-torsion point (0,0) of y^2 = x(x^2 + a x + b) maps it to y^2 = x(x^2 - 2a x + a^2 - 4b); this is the minimal model of that image. The 14 points listed are the images of curve#2177's generators under the isogeny, saturated with PARI (ellsaturation, primes up to 200) and LLL-reduced. PARI's 2-descent (ellrank) gives the upper bound 14, so the rank is exactly 14. Torsion Z/2Z. curve#2177 is the fibre (a,b,c,u) = (19,139,196,12) of Kihara's Z/2Z construction.

last edited by Steps Unbounded at 2026-09-26 19:53:55 UTC · history