curve #3471

y² + xy + y = x³ − 350118034100627899199059381753771143x + 79718109217596160364013393657928989116028766152608806

a-invariants: [1, 0, 1, -350118034100627899199059381753771143, 79718109217596160364013393657928989116028766152608806]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 44669319913005297558669849962700626503968806944547821921770
discriminant (Δ): 1427061502949408209948821118859574534012152857371141880883530382263634308365095837836356405097865102620500
Faltings height: 19.1551
naive height: 257.1443
primes of bad reduction: 2, 3, 5, 7, 11, 19, 53, 67, 71, 103, 113, 139, 251, 283, 293, 353, 439, 1021, 1103, 1201, 3467, 3607, 23311, 35977, 54539
regulator: 1441309252276395193898.852952436943026249415480591682880443113069876429
submitted by: Steps Unbounded
submitted at: 2026-09-27 06:36:13 UTC
last updated: 2026-09-27 06:36:13 UTC

Witness: 16 independent points log in to add more points →

(336723639306484325, 1939783285706651332429632)
(100304027013478586123/4, 1004286701102481817409869147839/8)
(-190796507246457898091087859949136755/1087765889089464649, 417970992072996486299306753849344584406146977250077376/1134496482463992637366989893)
(-253219474021160411940304675/1022784361, 12720000982820226661114418266563776737362/32709666649141)
(16739931735380598108644/49729, 24232619716782336362131771853778/11089567)
(2101245384445273844417531/122899396, 369970822943236918935776359647890353869/1362462704056)
(2514610358438468694305720/7219969, 97149647057384817261055857514143171/19400056703)
(3407488916351076119572730/619369, 6255043086629375944835529993033211071/487443403)
(7954951877440925879234888/21187609, 3365669916052609678348374467220437358/97526564227)
(32485240982384442001628165/144024001, 191087418367187361090986365418119567482/1728432036001)
(177312097848877669212260005/587383696, 557967831424152034813695806109342142147/14235831256256)
(2193963036111735666495800789/6326611600, 1304063546948965494669122698229757490533/503218686664000)
(26077237932572393807597078005/77502662449, 54445719968455985141417012868476328344074/21576198707164457)
(144174552310166095669917472675/180704908836, 42648911383786447365265859463928269676258753/76816572516730584)
(1752306117849194400534791843555/5326674745681, 146174601811236960044569850471651230606927998/12293746919367175079)
(152658079389260696386403812537064/430575572599225, 110355956242832188115528607056606599925667954183/8934578762739287505875)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=-921/2080. Found by sweeping t=a/b with |a|,b<=8192 using the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-27 06:36:13 UTC · history