curve #5330

y² + xy + y = x³ − x² − 35525801393692402802773166802132x + 105519834423546796918424091249830811877361922231

a-invariants: [1, -1, 1, -35525801393692402802773166802132, 105519834423546796918424091249830811877361922231]
c-invariants: [1705238466897235334533112006502321, -91169136941944424863945313802294816063036671547369]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 118911634774286608520500346082302161023536817904543003128504505533545853783719245670
discriminant (Δ): -1940540467105150355464865654179660645973810386840914409429030659416535252986878466528347136000000
Faltings height: 17.1851
naive height: 230.0736
primes of bad reduction: 2, 5, 13, 23, 41, 13093667, 5065697023, 1273551055088309, 17180854887289923367, 51411978794385181442027687
regulator: 268378835834197466063012.546073532080951070557232094052514265833979694114498015938689
submitted by: hdeping
submitted at: 2026-10-21 13:23:52 UTC
last updated: 2026-10-21 13:23:52 UTC

Witness: 23 independent points log in to add more points →

(19472919573827471449/1089, 81852866517314407020961127873/35937)
(-9680915988411038275/1849, 30586505953667061558492621079/79507)
(2627819121719097349/289, -3591112292723399375084831523/4913)
(4972523169902912589/169, -10881841930257278866163668047/2197)
(13869748172053381, -1510267593649995474278451)
(675796135691736809/49, -513231912393134735662307753/343)
(-174933683111300091/49, 148264681149397156067769947/343)
(-47451931761053793119/20736, -1248530309774951675679309393719/2985984)
(22540182353585309, 3279713816809025377259141)
(31672553325400489141/1521, -171873709967724961096474986229/59319)
(166753042862093779129/6561, 2100386384437228144896695786549/531441)
(-11520087584872810531/2809, -63554558956480159518079965367/148877)
(-2203941579133301081/484, 4427187325459845009115601017/10648)
(-7029991569323089, -88533841245488048378851)
(-73701663823368099/121, -474203718754523023841512361/1331)
(-3456897372619199259/961, 12874609640736672558876067059/29791)
(419577430762894042269/41209, -7479045446339265985478219570217/8365427)
(3776697713835641, 158802844398077608725829)
(4074831617548741, 168575528808112484387629)
(8859168458759941, 697209736364853202214829)
(26766265120243039/4, 3273410326945663582112807/8)
(332381580514018741/81, 123794864495365886795230181/729)
(8372672447080341, 628499827445905949988429)

Commentary

fiber 089 of the x1152 K3 fibration, E(u): y^2 = x^3 + A(u)*x + B(u), specialized at T = u = -68/89. A(u) = (-22234134433116285910092921277275)+(110922098753374475218610657758560)*u^1+(-2238835299613016412598155663300)*u^2+(-668291528103291204503696355619440)*u^3+(-1228382975540396813143327213163490)*u^4+(-1305046090974099330756097785833760)*u^5+(-730165036282222351438425309089220)*u^6+(-115184111329442799000859184518320)*u^7+(-6764206525058961744342207836427)*u^8 ; B(u) = (43437345573952011026007762008660031135091893750)+(-272653378639351264146660169730601346460528493600)*u^1+(274681422677639502079180012059258901947525880524)*u^2+(834095500002542547490222678572818893299399330960)*u^3+(-1446190337475684620303144196583106816058012416310)*u^4+(4817627385439859969516421772920670917055554020640)*u^5+(25741889274749849075903353261318279749128611700520)*u^6+(35618011375130752993917833852707386766033265770656)*u^7+(22305750380707309697813397419957100040991477108250)*u^8+(7478862206043393848507380663800830742000201885120)*u^9+(1781657943811020988023763350544021409036455304940)*u^10+(219381245128735196370337431665053397514385169040)*u^11+(9838320680474420154415220509114637635180818246)*u^12. Rank exactly 23: 23 independent witness points (17 from mod-p Shioda fake 2-descent section lift, 6 more from kappa(z*)-covers of the 20 shallowest 2*Lambda_17 coset classes, ratpoints H=1e8 on the p1 cluster, exact halving), canonical height Gram det != 0; Bober analytic rank bound 23 (kappa=2.8, GRH), rank is exactly 23.

last edited by hdeping at 2026-10-21 13:23:52 UTC · history