curve #4202

y² + xy = x³ − 1430648523620997724120540270604661759930x + 21226789013218564907802172151565455052046531129511330241252

a-invariants: [1, 0, 0, -1430648523620997724120540270604661759930, 21226789013218564907802172151565455052046531129511330241252]
c-invariants: [68671129133807890757785932989023764476641, -18339945707420840080444083432653265001104881795381324975156689]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 70478760662067086348938551878327869798723425742982584373150895811260297459676567219230
discriminant (Δ): -7245091667063850944674191560089578741141343902693415983553638991735353157509322457053223791216778190123906086400000000
Faltings height: 21.4037
naive height: 282.1284
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 43, 61, 361801483938649247, 52844667606505809421, 4673786882132408316677531614144453057
regulator: 491970832708754827634947.09006033095635740216685275378974552036818706683551488311737480
submitted by: apkorte
submitted at: 2026-10-13 01:24:38 UTC
last updated: 2026-10-13 01:24:38 UTC

Witness: 24 independent points log in to add more points →

(22354737381052709844, -20406999992942230009384358922)
(21609064501199951844, 20054962313619591220674495078)
(26842682846179204644, 46532704229340348497671991478)
(-43414134428823183276, 38868460106781345898242785718)
(2633925202921288344, 132200003136590609616850137078)
(-33361318507984853406, -178394817148106057016091448172)
(-11632452301562844156, 190511697010445639051249787078)
(22217881498367419614, 20206857822122601144130971288)
(8527937480749077294, -98216639228137601107725312822)
(-5321898557867458356, -169380711122085433794796489122)
(-17488884531624172456, 202232704336067538768076825778)
(26263727478469271694, -42058397638171676264471184822)
(29417537855547872544, 67810914355390737061026320778)
(29287033993594691844, 66691193435919876274544865078)
(24148390997367913344, -27585094319682916107337562922)
(14330249149351698444, 60564366181890770372952910878)
(-37799943146855256156, -145929138849936484309885934922)
(23712739726434900594, 25213583728889085845621797578)
(-1882579387245272838204/289, 854774193182178036919813366302294/4913)
(17502707309472664644, 39350087245903449014745613878)
(1531561177139084484, 137982809900246468360252427318)
(22503424492045964244, -20691175090828824111568920522)
(-42890549591067402156, 60717936835049254645136793078)
(826638510004407005844, 23742504170566595338193479257078)

Commentary

Specialization at T = -511/37 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -154503213541262535600T^8 + 4159635009352858651680T^7 - 21187086453270000393696T^6 - 687438318305734637863392T^5 - 4576470188855512350897360T^4 - 19322004287067422371639920T^3 - 39029689014386535692132184T^2 + 4186478995367154403195512T - 1679949255743694529867467, A6 = 747141710909816776767093360000T^12 - 29305404314392232038648503024000T^11 + 301598848438544320323778700280000T^10 + 2292679730804159433657382244285760T^9 - 4484131884783219751251582549192624T^8 + 544733850758018343321283421195576736T^7 + 7256453148985237089497289163544883552T^6 + 31881087967205708307154627785355135968T^5 + 53417637701365905114499013007709555776T^4 + 36362990355034819792007607408188452080T^3 + 65566855132366254216911881386291117144T^2 - 854195394262226327944905852404523432T + 424114741036883895390965869842474630. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-13 01:24:39 UTC · history