curve #3603

y² + xy + y = x³ − x² − 443614187006965842441597x + 98513667991088894052468455283418821

a-invariants: [1, -1, 1, -443614187006965842441597, 98513667991088894052468455283418821]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 47491363856755122892643998195872310287759933770
discriminant (Δ): 1394690915226111318270004923984400411925026433257830677990481927040000000
Faltings height: 12.5686
naive height: 174.9613
primes of bad reduction: 2, 5, 7, 11, 13, 17, 59, 223, 2358411632341, 817640650132716299516483
regulator: 639763390656740254.0140958548954135227327520103373697318349667275338471
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:08:54 UTC
last updated: 2026-09-29 19:08:54 UTC

Witness: 16 independent points log in to add more points →

(529821353311, 110471017586829144)
(617561381011, 245114989132044744)
(-410266707433800689/720801, 249792506699507065232080456/611960049)
(-6861127506103041/12769, 615125443693292122727368/1442897)
(-7992568932729/49, 139973190789346268768/343)
(136227274631319/529, 447431996486286655248/12167)
(541945286833549/2209, 6922795366023896631812/103823)
(2164851811514239/3249, 58196819593453278939592/185193)
(11858489024285131/64516, 2495152451345948153240711/16387064)
(3143726541054599/2209, 159141970013211920645912/103823)
(13912088661220439/11449, 1425307518991391758547992/1225043)
(46373964525388361/201601, 8414494998034104787294906/90518849)
(67928602103303551/1681, 17701917264632322609810504/68921)
(104706686953542711/10201, 33811557527305455773699944/1030301)
(493728647459937751/1879641, 27291877818327809227634584/2576987811)
(1010632146748927039/552049, 955079944848716360898242808/410172407)

Commentary

Specialization at T = 85/531 of the rank-11 fibration of X163, y^2 - (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #718 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:08:54 UTC · history