curve #485

y² + xy + y = x³ − x² − 74012522422036525660181051808532267783899782989795577x + 7697109057551217105862561100522932339919505723035449473880423044947230344156825

a-invariants: [1, -1, 1, -74012522422036525660181051808532267783899782989795577, 7697109057551217105862561100522932339919505723035449473880423044947230344156825]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 44679155598732028875449577292101762747529826674110173050234930184664207968832037401401166671113974
discriminant (Δ): 353453453909373287457091878916192430970772513287617772441739358270936631500110535138438668977455181501097630167554485167120265729618174723728707120689222582272
Faltings height: 29.2507
naive height: 376.8218
primes of bad reduction: 2, 3, 7, 11, 19, 23, 41, 43, 59, 73, 131, 149, 157, 197, 307, 353, 383, 421, 647, 1879, 2797, 12409, 12413, 21961, 1610579, 3714239, 8481911, 10050352438462415792051
regulator: 63983651922206937101.10448645432482410464221248850020524252241632009813
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:27 UTC
last updated: 2026-09-01 19:28:27 UTC

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

(-234275942970124510912958557, 3489724411080219976551251281282864684944)
(391145372188611490461314555, 6212135152830445883819602546473502485804)
(511820988010609383125556817/9, 51738721949485925070884081865527009105488/27)
(106410415896252114110458631, 1013069443600706181717430611416244532716)
(176128492795918295162824007, 353723526903114330324543823299663846060)
(7803457151684994457579943, 2668338564242129792631342380691975177612)
(1721819225497553629049060647/9, 19836145538164583088285145655535177045388/27)
(225803312245193533244173871, 1580474769582930398300200991907901751532)
(70112372409773525484242879573/529, 5652769379268774563073866099041136475201414/12167)
(92407249935493860683980803321/841, 23025903832875371818423129503644075481473974/24389)
(-4357155635666344430255252469/25, 488978657404993807850904487990298356950064/125)
(12483796532779087995866541412395/9604, 43210357726292792750981740111394847943882861705/941192)
(369796616907614443405609826294543/2569609, 668464898632400105111385708715598352510376014900/4119083227)
(-28766984952441690901365913, 3130881722809404512272630316369304766092)
(-268279912408772331775050949, 2871232102677307722168398394464190005292)
(729472481579286269108385026583/5329, 140664435845270249111254882333114890012650924/389017)

Commentary

Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-12533/7296 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 03:27:01 UTC · history