curve #3989

y² = x³ + x² − 265051851796941810370760x + 51817925876915587601587585228201908

a-invariants: [0, 1, 0, -265051851796941810370760, 51817925876915587601587585228201908]
c-invariants: [12722488886253206897796496, -44770687957731402621089192878553227456]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 11537416688626259646150573805118160
discriminant (Δ): 31753169925766772256447566567759566054536835266391447159701326400000000
Faltings height: 12.3294
naive height: 173.4162
primes of bad reduction: 2, 3, 5, 11, 17, 19, 59, 73, 101, 103, 107, 137, 179, 881, 1579, 1979, 2459
regulator: 255283205572818499.71321602499427030020491999974780444210948799
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:40:35 UTC
last updated: 2026-10-05 04:40:35 UTC

Witness: 12 independent points log in to add more points →

(783372909627574/2209, -5108976562518116085600/103823)
(521071371666499529612854/355653598689, -355443965010032551375080721698042400/212100069689362863)
(83827409094604305631/129709321, -572994714450660754189604777550/1477259456869)
(3919815516218786998294/1439823025, -241306046440834966932693372384672/54634084683625)
(12460616125791336026641/18626790400, -1060050231505868725477038448048761/2542184353792000)
(11141064930659991169/1567504, -37091912914529803923511235625/1962515008)
(18016473150249006406/25921, -76472369863840623525203419200/4173281)
(50585218077903615193/66064384, 293025298278168466013308719645/536971313152)
(1459716945670901951907766/14266152481, 1763589431331118932420784622796223200/1703963518483121)
(472433743803845706947974/750990360409, 238263581189273943519154971267602400/650805993359358173)
(1376696428091920575100022614/3782599697082649, -13883892619422722975062039239755878454400/232640635301521719471107)
(14778529530249273544922534136151591/2008488501088808618881, 1792295187055898467239884313539925841833845960344950/90012748833740728465299943585471)

Commentary

Unlisted rational-isogeny neighbor of curve#3634 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 12: 12 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 04:40:35 UTC · history