curve #5612

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y² = x³ − 8388560612127250143613319356191411672075

a-invariants: [0, 0, 0, 0, -8388560612127250143613319356191411672075]
c-invariants: [0, 7247716368877944124081907923749379684672800]
rank (lower bound): ≥ 8
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 33554242448509000574453277424765646688300
discriminant (Δ): -30398954029919728831809328673764177475483673242106857315877232567702028448636030000
Faltings height: 14.5059
naive height: 197.3785
primes of bad reduction: 2, 3, 5, 23, 29, 79, 83, 113, 181, 199, 251, 263
regulator: 9967446640175.87099581472711038136472141850767643814565
submitted by: Maksym Voznyy
submitted at: 2026-10-23 22:49:43 UTC
last updated: 2026-10-23 22:49:43 UTC

Witness: 8 independent points log in to add more points →

(5685757856539174026422717500939/20722023124034521, 13554838317690094592301022130402425441385766788/2982965503580553512385181)
(352626094616508483762027499/798571081, 6621733086864195785815179115880083982632/22566820177979)
(3871334758673763498046911276770094203714418628801/79985351574200726234587953090611856, -7329939368335958852169317505426152916758827532590463288895736732330959649/22621202481739151631139591381653196823209909128242496)
(77165642895543310733711330281/3163512623892544, 13924982805384760591421836463950172265042421/177932121844250013244928)
(1015061183053384999890095166615276925805297521736319775729/365674455719090006177370311282333164106896, 32339868168489594520530129899190926802292538539971180784832704395650500190325250572983/221127082243326627880151883059232204995325038053143398718090944)
(5730018293191304463130006424768236503060810587494369152344545468773440041/7960387269475337400401903283807761476214132023716253518400, 13716052179956781167707586615241407733389699633096181919323168912327252023434195958155724586975960190413189589/710233731684358888943848489037954022044830847512684435459197905853893607470003613248000)
(723312775101864257284081059514235358628061330182548619/16506089062587206593018452037219753234521, -583694759691043764242032555676355716022642907405188523187111252496634085573321828/2120636715090465743635882413858237651222414614792471193471069)
(33460060739753102011578732240875136499/1099027568878392239897041, 162251287789887754313465427229696569950582252852982374132/1152160229595934276252409019987387161)

Commentary

Voznyy (2023), A = -35252655507282684270, [r, s, t] = [46, 145, 263] for S2 in https://arxiv.org/pdf/math/0403116

last edited by Maksym Voznyy at 2026-10-23 22:51:33 UTC · history