curve #3834

y² + xy = x³ + 42459070568638227676679977396779231730x + 119881395284563761312108211303985196714695782262476368900

a-invariants: [1, 0, 0, 42459070568638227676679977396779231730, 119881395284563761312108211303985196714695782262476368900]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/6ℤ
conductor (N): 11562784654375022501788319763332764230
discriminant (Δ): -11107328478678772403490795196964524119060281489563172850238608426157217313471108866721347959826585379493933056000000
Faltings height: 20.7611
naive height: 271.7753
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 43, 67, 113, 419, 421, 677, 4639, 52081, 87421
regulator: 498533387.67651568063263206019866626901031222099685486300
submitted by: dujella
submitted at: 2026-10-03 06:59:59 UTC
last updated: 2026-10-03 06:59:59 UTC

Witness: 9 independent points log in to add more points →

(-31033395220068260, -10888696720875752056116741470)
(-1232864448860804240, -8103161016824343141105844670)
(-1328315624621856260, -7819120064970281438189301470)
(3802553855770780126, 18338947764785709323657418652)
(-2433590726687000900, -1463142233956860582883774430)
(3867651990684847612, 18491984028573822481136771362)
(-1218313370645706530, -8145220354674240158153680250)
(-1878910201911913560, -5785455130632454988224496970)
(28632932208035602735/4, 224941354774375071659805792365/8)

Commentary

Found within the two-parameter family by Dujella-Peral, y² = x³ + (1 + 6t − 3t²)x² − 16t³x, t = v(w² − 1)/((v + 1)(4v − w² + 1)), at v = 1/6 and w = 112741/62861. The curve was found with the assistance of Claude, based on our old PARI/GP program.

last edited by dujella at 2026-10-03 07:00:00 UTC · history