curve #4106

y² + xy + y = x³ − x² − 2778651284357309451088287825354625635003741447424457x + 56051416651443303183452415058912512169005454160909080801924823142470795940681

a-invariants: [1, -1, 1, -2778651284357309451088287825354625635003741447424457, 56051416651443303183452415058912512169005454160909080801924823142470795940681]
c-invariants: [133375261649150853652237815617022030480179589476373921, -48428423986847013950502886010711733092841870959955275536263910034286615049065969]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 374785364077444172381903170722631268547812541904984566738340877113861929593075923530
discriminant (Δ): 15795717957944696937904296088283719793002266325471102177991046678112183222968883342017914736253948595368690651508637229682452221093076510422594551808000000
Faltings height: 28.4230
naive height: 366.9750
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23, 31, 59, 73, 79, 193, 223, 389, 541, 643, 1097, 1151, 1447, 1831, 1913, 2273, 2591, 5227, 5399, 7853, 10601, 20020291, 161738747
regulator: 1140581778246779489389.29318331614777649063331955442162451232389726017503409135
submitted by: wgxli
submitted at: 2026-10-09 15:20:30 UTC
last updated: 2026-10-09 15:20:30 UTC

Witness: 20 independent points log in to add more points →

(28523511171921845194975991, 994395625273617755347975101220075404)
(96289531462073688202091, 236186076037392937935492134647347955704)
(28405834525268035741195841, 6475542387834145069288780756345504704)
(32312729696577623794173575, 1932298792862799653314410653364918972)
(20181457085593254453869879, 90520246786455283973896215709231746684)
(21356477500038598834812551, 80311168899957508016081479074079960764)
(26724664634371896826674591, 29662336355940001318739888147196282204)
(26905127113877085868505991, 27707252747407840277202691058645065404)
(27486063920207591243071671, 21035764226112213710173115920771804284)
(33018502943053657180615751, 17377976176565875700750753104179179964)
(33771633796500542584770161, 27002898718420095444877546522957054464)
(37078050517404814159287941, 63235070248144240167225324856364822304)
(39309693997266099125794439, 86987756978511561175421556687788420796)
(5220851981606745333045131, 204173445609574493810192906529410518044)
(7487977548436583704224731, 188851231610852939565751831128750807624)
(15824696111528417999264391, 126660731138930098212982421673685603004)
(16468611574340338744057991, 121480143193375664448395566024636681404)
(138099215919293107116808391, 1518573782592964817764026677734877684604)
(-203880596985233577419708881/4, 2043706067991933462372627016028114354187/8)
(434314721503563996453851968919/2809, 271375222058487668113328014166385229848442308/148877)

Commentary

specialization at T = 245484/7853 of the rank-9 fibration of X720 given by: y^2 + (1260T^2 + 16274T + 6718)xy = x(x + (2431 - 50T)(81T - 416)(98T + 405)T)(x + 8(512T - 4465)(2T - 187)(361T + 1924)).

last edited by wgxli at 2026-10-09 15:20:30 UTC · history