curve #481

y² = x³ − 20184339727045085994929693932333023735201867236285367x + 1289020600247886909318904446129761341665487540840774607412800550174597850870970

a-invariants: [0, 0, 0, -20184339727045085994929693932333023735201867236285367, 1289020600247886909318904446129761341665487540840774607412800550174597850870970]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 53806326363419427086537383779816765270954899278777110096654970448126165901027556398264
discriminant (Δ): -191511835100981240213910393778202367287469014388481494957834096697504959138311360261067989515132397137021293676057567881393449591491070870462397925022622413568
Faltings height: 29.0931
naive height: 373.2342
primes of bad reduction: 2, 3, 7, 11, 17, 29, 37, 43, 61, 67, 149, 163, 167, 223, 227, 311, 389, 587, 601, 631, 641, 743, 1361, 1531, 15823, 39367, 57139, 920478348419602499
regulator: 11468189097092191091236.255181345108355748190878849199545439131484346729377
submitted by: Matthias Breddin
submitted at: 2026-09-01 19:28:11 UTC
last updated: 2026-09-01 19:28:11 UTC

Witness: 18 independent points log in to add more points →

(78191787901563442255322347, 434548174935391497444659139411609474612)
(143055757596403498547188477, 1152893526816062274200116846826649628338)
(-98920596219647664262558031, 1522399867849516516722393916601157701166)
(267402466093412724682286503, 3874538089515156732347288035192804873836)
(-104411673027215270227132567, 1502739620773518016048915322990245498214)
(83735007366667217321824363, 431273784918713846351364160033198066164)
(11862090209484476985648003349/121, 668738809914619807876950550711918938571254/1331)
(34972584424436046526611913, 791136146153712298186782683040649958586)
(7043630756888011703774999851, 591026986529626004846790191013107172395052)
(-719786985295388179010364667397/5041, 401744682537322825775227552073405162495652204/357911)
(-11893526415054905240655827, 1235880725490616504474883040658189532514)
(102049740626531120726751842119/961, 17449845470243871661915218010393304327634564/29791)
(1088809519885090076951414741896809557/12881569009, 631933430195019776432558103353887228405951034856837622/1462019437814473)
(10528267111640324697474302045689/128881, 19916879575123245781027330125553296965752426314/46268279)
(87336220579029582332450086737577/130321, 799425468789833528909169395563867898860270783578/47045881)
(903802262322084526735085198446789/4116841, 22782340518748659936359216570214096808201018521894/8353070389)
(-1432979789658835215103498221291313223/17873283481, 3695565829683973370018101306396926342466100971758717798/2389497141858371)
(-38059358612552434242604427474977074191/319933903129, 256348224155215917248053062184660247707553817967842509694/180963253825146883)

Commentary

Exact rank 18, unconditional: rational 2-torsion point, 18 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-3259/10943 on branch u=22/17 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Listed as 'Breddin (2026)' in Dujella's Z/2Z table. 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 03:26:59 UTC · history