curve #6437

y² + xy + y = x³ − 42307864406752182296988420965x + 3349589141764027524569965404051711395908992

a-invariants: [1, 0, 1, -42307864406752182296988420965, 3349589141764027524569965404051711395908992]
c-invariants: [2030777491524104750255444206297, -2894045018484122827394687395257804029231678749]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 143404793018081973616765790502250200794
discriminant (Δ): -262727265585746560745305119108160496400999117811645133581949225065318043073250193776
Faltings height: 15.1029
naive height: 209.3580
primes of bad reduction: 2, 11, 31, 67, 103, 131, 271, 349, 601, 3163, 6679, 10069, 6024637
regulator: 19349747903205.5663621036045540308400783302687108669958890816
submitted by: Maksym Voznyy
submitted at: 2026-11-03 04:34:54 UTC
last updated: 2026-11-03 04:34:54 UTC

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

(4537772888289827675315456265/38030704945216, 3352814559131065464712126009534372610487/234531706449472331264)
(3945224115069458560/33489, 124097174918179272953834521/6128487)
(-24959491794559668832/113569, -54564631148807883826854753440/38272753)
(51552167053679035096/289, -370143477645643500111454952273/4913)
(225834499799283868035/1570009, 963984760734464930077557721866/1967221277)
(119574724983962113039/881721, -269764390976755297801898929534/827936019)
(426135384159409199388256111/3585641429241, 65723517043249284044332409034126193154/6789695311940743539)
(346157091803402994174384/3062404921, -18221232676133545915622609858059344/169470425923219)
(35104759486535938363223/301265449, -225004030163890186035452036009074/5229064398293)
(24971818700143105255/210681, 1009011820797620222674849534/96702579)
(43809960792356754343951/256736529, 4312412702949836209957107970775962/4113689404167)

Commentary

Voznyy (2026), T = -1094/737 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-11-03 04:34:54 UTC · history