curve #3194

y² + xy = x³ − 8620423182399952221292535451x + 307993925882250651596378357172269498313681

a-invariants: [1, 0, 0, -8620423182399952221292535451, 307993925882250651596378357172269498313681]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 37047639080308908544920171799389911308078271310
discriminant (Δ): 18657376427921023421562941463757361799811320362964128307693739144084575894896640000
Faltings height: 14.7708
naive height: 204.5854
primes of bad reduction: 2, 3, 5, 7, 11, 29, 109, 113, 157, 173, 541, 827, 2083, 4231, 5171, 14177, 1708781
regulator: 1821088857816859.24352758318231258467301407176464633221833703086490
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:04 UTC
last updated: 2026-09-26 20:42:04 UTC

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

(29093173328111017326/534361, 2589793618487742939741679347/390617891)
(2897923327950033486/44521, 1414172828802726748508404827/9393931)
(4957470356280809506/77841, 2856686448349795537666708163/21717639)
(3365457178734911624286/829921, 195188056946779226073953255873727/756058031)
(329859401341993576014/223729, 5979314538835318941954862494489/105823817)
(95750181892299249474/966289, 620512883011027916764823306979/949862087)
(-1264828292670667338725769762/119250964603681, 821790920259739013574046578569633316149025/1302245456923798689329)
(5359430132780313/256, 1514404970292049978539675/4096)
(8772248890687182/169, 43312652419056610859589/2197)
(11306018566339271922371834334/238943797249249, 287052924139605892576319097109903152538431/3693547101726021936943)
(-217190686146220834/2209, 47018791854231809432911519/103823)
(108059385100640211534/1974025, -32679481753615930720539642987/2773505125)
(97699884711223398068876287716606/12254654943555025, -965632242633271672550174172878308007990557511031/1356599431969474215993625)
(1356594417467846/25, -15734186252882024771/125)

Commentary

2-isogenous image of D-z2old141516171819-072, rational kernel x=-423545465356529/4. 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 20:42:04 UTC · history