curve #487

y² + xy = x³ − 16229788060744054639441032025869941478736582150x + 2231172574677504566226588409051731091267083694718472662914036595001956

a-invariants: [1, 0, 0, -16229788060744054639441032025869941478736582150, 2231172574677504566226588409051731091267083694718472662914036595001956]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 1153464240454909671490191378607928470096426970591096070526119088776917088138
discriminant (Δ): -1876951088311557689801031867173545551996530224234226596557815816398336270726531377514044949885697047169521540877941918663544361139837710532608
Faltings height: 25.7932
naive height: 332.8849
primes of bad reduction: 2, 3, 7, 11, 13, 23, 31, 41, 43, 53, 61, 71, 73, 83, 101, 503, 2371, 4597, 4943, 33503, 55661, 175261, 385709, 1235239, 1657021, 5017709
regulator: 2485704261577274679.896116864024013223100454186381227638500724045938413
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:35 UTC
last updated: 2026-09-01 19:28:35 UTC

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

(16734849667799038004684, 44319929974000082826896938646874890)
(-11540026265682030394384, 49162261384398890643734381027359046)
(237040688947921347490820, 108180233781379500625245716140578506)
(1329008925423939652145708, 1525792653688211206948273475931301418)
(312046144831912554815948, 165986552285268789436876089909462794)
(-12677641695845200071426508/169, 120865385457740551862780690677237008338/2197)
(446404395463549416141164, 289731256352779343849619321868673450)
(1178904038299649308852214, 1273403317635057139281402416392777550)
(232680361350911431110866, 105129298578224661799360170991900046)
(3103921437861916789052917052/52441, 461668347262127852240934271623975527597906/12008989)
(-143970089360107455970516, 39795136721842277254497573970504490)
(-103259420461270604948217220/2809, 7846973801082774387899550688501994872162/148877)
(138768535940101935698007438380/128881, 51376958746620981034928717392317765376502854/46268279)
(932659834078475926859945173557524/11098411801, 44686789127263894494013135478584502138177622171194/1169206584823549)
(-1031377801517774091392648/9, 1373051092654952213040309794293592426/27)
(1241047655482511415294226546388/6620329, 1294628634489126547884397533150710523876427858/17034106517)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-5259/12143 on branch u=19/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Preprint (proof, measurements, decided slices): https://doi.org/10.5281/zenodo.22242107 — data and code incl. the per-fiber ellrank certificate and witness points for this curve: https://doi.org/10.5281/zenodo.22242109 (Breddin, 2 Sep 2026).

last edited by Matthias Breddin at 2026-09-02 03:27:02 UTC · history