curve #1992

y² + xy = x³ − 17793535500792068066953133544289432544163133505309985x + 3425836650466723498778158340740131327136083787001388794367970551929025190749897

a-invariants: [1, 0, 0, -17793535500792068066953133544289432544163133505309985, 3425836650466723498778158340740131327136083787001388794367970551929025190749897]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 2707707219640613573728887421621760566601621577410675550720868877255002084906983107620003530 ☆ record for ℤ/2ℤ torsion, rank ≥ 20
discriminant (Δ): -4709555104341233533154426589526443049767652747680412194363550872103014773707457702882775417624789495709485029069326115467147471064178479825350451373656250000000
Faltings height: 29.3214
naive height: 375.1891
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 59, 89, 137, 167, 181, 239, 311, 373, 379, 433, 809, 1021, 1123, 1367, 1543, 3359, 301643, 458483, 60388687, 332302241, 27570707443
regulator: 7560302866322941965532629.66115725558939743096130075649016042546625301754205367
submitted by: Steps Unbounded
submitted at: 2026-09-26 04:29:17 UTC
last updated: 2026-09-26 04:29:17 UTC

Witness: 20 independent points log in to add more points →

(-183514638897182658602180466, 714754591316577888381903676413947683983)
(-181490008730746412062336716, 822894002391080888427669534434902371483)
(-1833952910750150661503340771848535749409504/29652930577269169, -10575898685033758711533139863176778194134436882944680938362563099/5106242444126492723591447)
(-87436556479688315492529066, -2076818819561263556676188508139245743817)
(158367329715926105170059534, 2140048600149243024161777578131470016483)
(-6794469876274935059468882842597996941914/36221415406929, 88058565454017127482916343470443587708880429952493360288561/217995799577615852967)
(52433925679136331420855684, 1623887062802066951024954890751632186683)
(-188706547275275976908173566, 252433696896813235516444165411246957683)
(11781786402944193630578709534, 1278760861076431930382782783600584983653983)
(31090304786107634536167384, 1703726009770983436880525626585688128083)
(34321335740845289783873184, 1689842508116004467004701205160096475133)
(-26822777265609312296855766163664607610789112272260291/159246224045583815737234576, 2576858389417759583861747502283537040096267424152883884209805407007973042884133/2009572666999807426247030338879955562176)
(2286721056219002898593078048514/3721, 3401559422898301896595274033258761339008707823/226981)
(-3245261213509975193500089977329211167986/24379409877601, -223121479048197073284639558247018572022879489561788492238527/120374579620558695151)
(-47415780303891394703168511291/400, -15736639468783105464381136336028516898587763/8000)
(-695742814226650545358537434/49, -657596528739552913130758954427414485114431/343)
(-765452678620222700082666714/25, 248178914683803965897772118237790973636519/125)
(5225247747741890130153438632916/11449, 11645604577695067591882451299322066143717350819/1225043)
(277601056838902874159793443972916/368449, 4570928127751873056670373513242249566268093449669/223648543)
(-229470299930267569188291019177286334/26489841049, 8156795315076991530580458554325416796389998864620293919/4311407059612093)

Commentary

New curve obtained as the 2-isogenous partner of curve#802; the 20 witness points are transported through the rational isogeny and certified independent by the exact 2-descent character test.

last edited by Steps Unbounded at 2026-09-26 04:29:18 UTC · history