curve #505

y² + xy + y = x³ − 2497797449771471795848514548362123698527267551x + 46349171130280580414429905307506113953211289710155681032227157688998

a-invariants: [1, 0, 1, -2497797449771471795848514548362123698527267551, 46349171130280580414429905307506113953211289710155681032227157688998]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 114713100997610410312595038097570863411319285752217220984538901104669511816895282450
discriminant (Δ): 69317140582662900877610095614907960786971423401250978324706785878258678236968916883842023737085086042782754420373162513987494823429937500
Faltings height: 25.0235
naive height: 325.2088
primes of bad reduction: 2, 3, 5, 7, 11, 23, 83, 113, 131, 139, 157, 181, 383, 461, 569, 1721, 2381, 2521, 4049, 8117, 13697, 226769, 340757, 522157, 844717, 1973893, 2830897
regulator: 55619160923131104523648063.499729584626965197303552507589585354316886444213
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:41:21 UTC
last updated: 2026-09-02 13:41:21 UTC

Witness: 18 independent points log in to add more points →

(8046669332028274407152, 5174092565810755779402316655974386)
(29257586126160441216460713169556623607/115159275850441, 155390824282337764874406193474544069819287286321454494454/1235800560625252121989)
(1441253172829817864212936489134578/63704245609, 19152400571715159357184775609442080363128196656368/16078760478974773)
(101220127608732761787752, 28819681688301943479648893491860186)
(836591976494771131026464508038372/157616146081, 360776071120953020414939547829914112499509728825774/62575028539471729)
(1822493742923123957768393356583/84309124, 1213184373681653061457104454939517513089161963/774126376568)
(24314347157433471439372633160245529713463402/1739970703576819081, 119892259737335277923238501083081162081409462462074784679090172136/2295159670029022392764567771)
(794407329000130996778859332868744316493/32913863346899089, 2091762563822303042719868498660068517460134716962646068972/5971291773812780247455687)
(62657284861968231220003068282128947025999258/6784820729711320423009, 2741850574090631833310915239735762720701378879662014653893636452488/558865841233048966842317203690223)
(1891309600024177268569432701077/54213769, 515631816725770949394711599240774547884258538/399175981147)
(2118433284028401613445352877372664912/39758687091601, 2013846656180349863279597856587343270869021004174423669/250696373763048433849)
(21619782409441843477285825679597/203889841, 90847939897635729536697785816632438462072065324/2911343039639)
(18521404772795566261228124593/762129, 1977182377523897666991056927568125412722/665338617)
(1280419055380765168940936457593/67683529, 1348729589974914016870000821797547058236867828/556832393083)
(-11545597503491542763505183374325935050242967/1556965291338350647009, 493260107455945210891428021649834210691162014433441666836663047012/61435358477933425374779654036527)
(17776404705665020708952837063963/309126724, 52378372398205007287759007712653285800037558733/5435066061368)
(69114581985415856170174396536557271350071/7041437300684248900, 89176185172926351249824561033071681148959936526635603966061659/18684951495739693812019763000)
(146026031744432562803108110311840498164399177/2214722655764177523721, 1352122879165930910261535498157379352674194672740981740055833245014/104226708412194954173815356693019)

Commentary

Exact rank 18, unconditional: rational 2-torsion point, 18 independent points (Neron-Tate height matrix of rank 18; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 18 (PARI/GP 2.15.4 ellrank, effort 3). Fiber t=-10949/24231 on branch u=29/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:41:22 UTC · history