curve #3977

y² + xy = x³ − 89712743925534611792099041419519x − 202120810012069252211057582477002830047053025616

a-invariants: [1, 0, 0, -89712743925534611792099041419519, -202120810012069252211057582477002830047053025616]
c-invariants: [4306211708425661366020753988136913, 174632379850427827451036188621638396129522831926855]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1319483920430920599678754550348754589797
discriminant (Δ): 28562264678516369242872664169896291489456887940749513934695788509892290354472895131671025763221649
Faltings height: 17.4009
naive height: 232.3361
primes of bad reduction: 3, 19, 61, 71, 97, 167, 229, 269, 443, 449, 773, 1229, 1823, 1997, 2731, 2851
regulator: 79823936291445081.6354376127476915163320581662648782941578244
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:38:08 UTC
last updated: 2026-10-05 04:38:08 UTC

Witness: 11 independent points log in to add more points →

(-45167134881508859734212/7038409, 6173909876107188868171022166702318/18672899077)
(12391172860869862430193888/1203409, 43618270304021154152724715092325766652/1320139673)
(-90389865752547635664385671/28927206400, 1074507868951337225960725381991185990173/4919939264512000)
(-487224045725521773972973167/62244262144, -2223766067821361037535280357717811206817/15529196473782272)
(30793926678244902447657/2483776, 3010125930241612446546242500062249/3914430976)
(4237395860559295484453062563/272887132996, 208970492208231121385142173454188598059031/142552417857248456)
(843959009222994276005433445301244/48959844237751225, 19897535010767064413985924325613657548000408306842/10833281821087368045425875)
(97310457949945858043393524463244/7049805016355689, 645575979471670847361970680647545453898321773986/591923603415459629122613)
(654112810099904985281315380820307439308/4472683226456764196761, -16693682173723707237783527688953999622232043283942668284974/299124655295884767664888482455741)
(-205848566783670426806250417579483/84970218840138256, 782788897533458303888524954144154328343028085429/24768522755657862582316096)
(-12044804644761738691114914679623/4709931954555136, -33246097684155456758089416730593240351448924149/323237656352974973456384)

Commentary

Unlisted rational-isogeny neighbor of curve#3702 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 11: 11 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:38:08 UTC · history