curve #4263

y² + xy + y = x³ − x² − 337990727551236774111124343499960380x + 77277759626177159577984375591439095158720118318203647

a-invariants: [1, -1, 1, -337990727551236774111124343499960380, 77277759626177159577984375591439095158720118318203647]
c-invariants: [16223554922459365157333968487998098225, -66767984317017065802372503359936235009131324030936509225]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 77519333444665700271122771572822177358597129155530657165098398935235427803771301026650
discriminant (Δ): -108717297562017607394043209883866872004269333051384218482529289499686823652017744244499301325689310904320000
Faltings height: 19.3222
naive height: 257.0816
primes of bad reduction: 2, 3, 5, 7, 11, 109, 197, 848123011, 206803331064923, 716684121495967, 25601293634714065871, 32374867301916089857
regulator: 1777583406172718730373497.0779813197637155148641451437876384172228427491977002449397897
submitted by: apkorte
submitted at: 2026-10-15 21:44:36 UTC
last updated: 2026-10-15 21:44:36 UTC

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

(17478062179079125, 267162265139180935401961733)
(1018212826255971109, -888127754207422011244053835)
(632875145679948949, -341844865762129013718761755)
(1107545878836766069, 1030299662181403586864681285)
(318602999334585589, -43972858795990064222646715)
(-351487090999232411, 390708722877737486691945125)
(-25875995514392518939/49, 112981026363718723363831021555/343)
(593930493853461685, -293335609608744108975749179)
(1681185062906077429, 2064152619828313140331767365)
(607483661948006005, -310060475259445460683200187)
(5776559597602186093, 13815961648839001622644463285)
(143871472179837790117/289, 884250298267314769009128273925/4913)
(2187366667685300725, 3131067464460896687902859333)
(-300644818931488331, 389510520058860602388027845)
(1171030211245984405, -1134604734158640946328010619)
(-20725545538692936899/169, 751114754893092927436287454505/2197)
(-235642587232727051, 379259992885930589254900805)
(4246658149926240941/25, -19671495922607921731875537679/125)
(337687908761050625, -40620004334562934110761359)
(765010284783190069, 516165243379232361679760645)
(-140125422682858451, -349123862942048283920705995)
(-463948842080620427, 366365747745135501344183621)
(3786612438678977069, -7286393451068601640443985355)
(58761314173122251965/169, 93046984557580181142783943721/2197)

Commentary

Specialization at T = 1154/167 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -4765105671341170713012T^8 + 114715794415427693698302T^7 - 1289703173484205490556069T^6 + 1970946774680673720492576T^5 - 1710136570699445082168780T^4 - 705937567588433468644896T^3 + 1908320361802866590386896T^2 + 127404060867858778875648T - 648856125950076308077632, A6 = 96897205978949562584426677136484T^12 - 5182597562294583468083790262057896T^11 + 84877162983356922665937799396845291T^10 - 478870362976518748392317900380853010T^9 + 1861233683275843787028761356431736485T^8 - 1364843279709253439363789384039551872T^7 - 1311792976812619745745993240617168352T^6 + 735145326496711370457296224143481152T^5 + 1094740667609718961081479717092760480T^4 + 53782848467549070290540339428638720T^3 - 759905051149062814344760081603028736T^2 - 65476590517227188924250720852312576T + 200491211946380337902432377627289856. 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 2026-10-15 21:44:36 UTC · history