curve #4260

y² = x³ + x² − 29624506444875281145047505156889285929565x + 1956876867943529679986667245509071799032032313211123512714404

a-invariants: [0, 1, 0, -29624506444875281145047505156889285929565, 1956876867943529679986667245509071799032032313211123512714404]
c-invariants: [1421976309354013494962280247530685724619136, -1690741613903209643517012357975962115333449600099594829332959840]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 4886783027910672419795090416419690625331967682390524060138371237024089995718528240155263527122564282137268
discriminant (Δ): 9636881933818402648783263637967977915169408963573305335816523555453964261278867945723818143510532247613503906400924361552
Faltings height: 22.0790
naive height: 291.1819
primes of bad reduction: 2, 3, 7, 17, 19, 53, 193, 19053677491, 98063700001362749, 2684220103476677117874079, 3510773890401645166283691045154686287893234271
regulator: 88194201700180040915486499051.1163640391738168489234898881755818253483109700788060636398885
submitted by: apkorte
submitted at: 2026-10-15 20:57:27 UTC
last updated: 2026-10-15 20:57:27 UTC

Witness: 26 independent points log in to add more points →

(90829701228403989647, 124257710630847592898882337441)
(122692244077422510095, -411234111774782959054547564127)
(104325104447446946195, -41750910591119224731818009127)
(849473790054888256121, -24285320277589204635166518392829)
(-157621364256347471023, 1646301575222467612733870960931)
(63857764810198878644, 570545526844000432167937131708)
(94813063219603150841, -20255989281505732231585939971)
(-198519366372097277221, -119494904527523911764682662057)
(80357879445615256064, -308571580431511063318021826022)
(73805923928135468747, 415279269398015545890361103991)
(126058854813426388769, 475001008101442711309128421917)
(109781620386197337731, -166550576326468997972294199879)
(488631603808501832789, 10205268297288461884614016954797)
(34388011827212712698081/64, -6083199322619996950050195276351759/512)
(83677264583454036788, -252740489320694985931310709300)
(485032727434514065775, -10084405941794426322935476722273)
(15943256345579691725483/169, 91481056921740714399834635919987/2197)
(118626115327449467897, -334607023276474951946231149059)
(976358315603888844233/16, -39262298315884005293675075371725/64)
(76017004857848338325, 379712185376585809836200857773)
(364579328652732464909083991/4713241, 3663486896018276853723974376247293028131/10232446211)
(208101934272216893619668/169, -94050614094934858557943220685627948/2197)
(780532836961816048529, 21315702553388941379476153526307)
(92104444997353137863, 98342375642441753195160383535)
(83564466989298807713, -254662783498086565275890576355)
(94672450082350951697, -28096058696792610050211418659)

Commentary

Specialization at T = 458/1105 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -1104083919542990740672T^8 - 10725368859924614083328T^7 + 27233182276536009979376T^6 + 206542082020468176713536T^5 - 397181427596358623750980T^4 + 4843779804480200729063344T^3 - 16606432570794981193216279T^2 - 25584496459329575271150212T - 11348709459926316525508132, A6 = 12350516786472668766297337795840T^12 + 221799272750803971404191627025920T^11 + 569212223298482978503626062159360T^10 - 11869075609185296079185311651020800T^9 - 53903362742072871161532361492760160T^8 + 383972047913208585082616093455327680T^7 + 1230371364926344316718891951621143520T^6 - 7765129801662673724018170759694598720T^5 - 1399242604874728526040920920131327735T^4 + 17367237161504277326036112974588085430T^3 + 57428692355319025444778734345548537226T^2 + 49825188772368199520084299618572523256T + 14830879968852803906439396097627134769. 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 20:57:27 UTC · history