curve #3976

y² + xy = x³ − x² − 978533862463073546815824276647319x − 5825826861087317454214787039594217276271607258675

a-invariants: [1, -1, 0, -978533862463073546815824276647319, -5825826861087317454214787039594217276271607258675]
c-invariants: [46969625398227530247159565279071321, 5033514407979442491804890294233289838916712427316131]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 38689313102431608020253080619974416595190
discriminant (Δ): 45304149467366972609069043778706862859891739649150272251036561230001654669539751780208940822362250000
Faltings height: 18.0101
naive height: 239.5044
primes of bad reduction: 2, 3, 5, 19, 23, 37, 113, 139, 211, 281, 313, 431, 547, 2029, 3919, 133319, 364943
regulator: 111018110862519451.660588693018186992864675730123786685921856
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:37:57 UTC
last updated: 2026-10-05 04:37:57 UTC

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

(4287477770213909005729896329/28861932544, 274188796430468155565769448237671541617749/4903295996035072)
(-34023926040963483726538919831369/1260659774304441, 43046912097990714801747659279780091596078286662/44760697255365166429011)
(-425939181099764684405424187254/52318618051921, 478838111455908094075546642926535942484505457/378428987667050952281)
(-55179383261338813255895425/8512092121, 388784526006554251597700523966518531770/785334131175581)
(-20532701055641983527850/919681, -1950852842203966960805604698531765/881974079)
(1596198668164577306854582957079/6244936026169, 2001135169879988920225650727513438422269329501/15606013945227990803)
(-16488415577637983647150929/2098464481, -112834488210175081679838710046005688257/96128559410129)
(-2769184403528955810269206/445336609, -1276338995570793079406642371406735341/9397938459727)
(176369742972291943480024798271159/4261327343418649, -1378936493625006706182125561938996647697619277331/278174518622633071343107)
(1247015834694188737193946689078839/26372479329147249, 31362715923454357464876950775826272805955964274511/4282786656367518409997193)
(3122985219344036401956145361227896135071/53802306176946508166881, 143893213275922446226940586398338507319561025143889533248318/12479619310332089970464813243231279)

Commentary

Unlisted rational-isogeny neighbor of curve#3703 (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 04:37:57 UTC · history