curve #4190

y² + xy + y = x³ − x² − 4169027949887435764264668831464858623682x + 103324512130810530988859852669996695347256952511610914713281

a-invariants: [1, -1, 1, -4169027949887435764264668831464858623682, 103324512130810530988859852669996695347256952511610914713281]
c-invariants: [200113341594596916684704103910313213936721, -89272378481020298773474402669701458654948838502435420849559769]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 34095396583563628150779759927351282847188652317457175606079208548621263015281621315321866798319264290
discriminant (Δ): 25492558413978830582805881090534313419836148855039646459275152906235514735546453673290015135449774521166822252544000000
Faltings height: 21.5867
naive height: 285.2991
primes of bad reduction: 2, 3, 5, 7, 13, 19, 219108004521326573811321636960036519807137409661700248095104482672201420315414313445934495201589
regulator: 10944798641777078125716155629.6778311094694256496234578874497567105836202929354092421571516
submitted by: apkorte
submitted at: 2026-10-12 02:25:24 UTC
last updated: 2026-10-12 02:25:24 UTC

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

(38898666444758816641, 3563687595412192784915023679)
(-10488295476422635759, -381964332198825882629558826721)
(44748894234164403841, -79832012934379391031687510721)
(39082673948184958225, -9208342323815267821459522273)
(-26317644147354070559, -441379000107090317560372583521)
(5309563614594197031889/529, -3041535569141708336932952320403207/12167)
(33032343783405200641, 40675771489826702441463679679)
(-54773439009567388799, -409083235341753093423590996161)
(30584674567866615841, -66526206968858669200578201121)
(39574729690036550853601/625, 4770486496761725377263249430436951/15625)
(-17224341916186487039, -412338686596758624774204490561)
(39856756869875559041, -21805297908414229733326729921)
(-5109897596476104389, -352837630108427477051290413001)
(44076213462497314561, 72089952358653091235755314239)
(-13017472925119427519, -394193924233001726347731631681)
(-73503477045242835275, -112431459428540213055082712473)
(29826227195300877601, -74240576674138569910612457761)
(39933943847335728961, -22852524406785289975830086401)
(40937124626534006641, -35509166094292035386652346321)
(520237735315530503041, 11778598300158063755601678722879)
(-12053907688218379199, 389648812775922685279675261439)
(41231632789549386241, 39043803693929523743456825279)
(18244399543114976561, -182581206441222818952791849761)
(41314663304900466641, -40032272917678252234676926321)
(114073442349056698369, -1054589189272992782788084077121)
(86125544733623322463369/9, -25274822382685945185277073103371867/27)

Commentary

Specialization at T = -373/1268 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -73932522356183031905227T^8 + 649309903876882877687472T^7 - 1685916835628140824312084T^6 + 645581139407117375333216T^5 - 406054368862523371209330T^4 - 858600221311968331297296T^3 + 1074013049265607843957676T^2 + 189439686366819237655008T - 279637802575984309948587, A6 = 7677120880090165783883135437481350T^12 - 104661205483741457154365858257793680T^11 + 560585458424820914639770163531974620T^10 - 1380712030244078852090340039520473600T^9 + 959024595422485011830600330788665450T^8 + 2176075475403505129770137846307997920T^7 - 1812178037587130500705886817575824920T^6 - 1469296539009918506803974573439080480T^5 + 1256479376113039680818470745801469450T^4 + 491406111407235947158204600615297520T^3 - 440604154329209614719459225390266084T^2 - 63212663150279697924996150457217184T + 62214320185228162206822083588855334. 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 2026-10-12 02:25:24 UTC · history