curve #3186

y² + xy + y = x³ − 23869924141198235915266828x + 73082734473257982724331044652051853696506

a-invariants: [1, 0, 1, -23869924141198235915266828, 73082734473257982724331044652051853696506]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 160217014633589313145585973576996191511802644586870
discriminant (Δ): -2307348315305855930627187360038199656477517988388171855018247560817219758044844550000
Faltings height: 14.8667
naive height: 201.7080
primes of bad reduction: 2, 3, 5, 31, 43, 109, 113, 223, 233, 281, 467, 3109, 5441, 102031, 1148507, 29485399
regulator: 1564774134187081.11984933526168249032116397704098685075970553360262
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:39 UTC
last updated: 2026-09-26 20:41:39 UTC

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

(9940242686605, -271712428968492642253)
(-26965344324935, -232635086872892992813)
(11605656304330, 272706592555483007747)
(44661778467730, 401375574677567219522)
(683782069570, -270308587617340913893)
(-41848547446470, 28147069457527213747)
(2374225514355970/49, 147801962771457380557646/343)
(1841085442195645/49, -121380112054247265117079/343)
(9149470611795778/169, -1054725936903635784048409/2197)
(31868761274617056745/304704, -185373621648198605571111350449/168196608)
(1214258078975038152447057333505/86466713379868441, -6987078182803909502642779691949024026684516417/25425729803654438802804989)
(168883209147820824047318005/789176382736, -2202321199230670788231179385915129303923/701069574661822016)
(2419843639400480409510441143350365/5487431991435331921, -119080134318259752346036828529840615827138569938407/12854456717124483157815927719)
(-1410238896503604387126613809808807655/35788553153105271182689, -767129465301277749679284916572818177942042305791460861/6770429370475130134719407846645137)

Commentary

2-isogenous image of D-z2old141516171819-060, rational kernel x=335995856219831/4. 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 20:41:39 UTC · history