curve #4607

y² + xy = x³ − 49370170918088587382898816117738443320x + 131084012722641939865086973397985087435670268435293109312

a-invariants: [1, 0, 0, -49370170918088587382898816117738443320, 131084012722641939865086973397985087435670268435293109312]
c-invariants: [2369768204068252194379143173651445279361, -113256586992362636046989797321961493835987826688570414364609]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 38938275751421165657965390788800553681436254990327719933556824785334970630362562521910
discriminant (Δ): 278410278436078887242027930949421365246835245842724473608561432853047376426476257767241141416653559867418624000000
Faltings height: 20.5578
naive height: 271.9908
primes of bad reduction: 2, 3, 5, 7, 11, 23, 29, 41, 71, 1051, 21405158469588865084313073193, 385900670961641322828141340373091638822278871
regulator: 706079748940218750438424.97259291173698926205429528727837066422782961769642587458733802
submitted by: apkorte
submitted at: 2026-10-19 01:29:06 UTC
last updated: 2026-10-19 01:29:06 UTC

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

(6520550819144364464, -9295233451784892691047723832)
(996154257209098064, 9104516124338530397011092968)
(6220003627532189264, 8040126833622198126988282568)
(586206676512376304, 10116537547537493381517585608)
(-3617390020297312336, 16196906527305460220697893768)
(-1480752900039860986, -14175411616247854506053217082)
(281192301017510808464, 4713796091648297372967733268168)
(4737920360026823024, -1878400035844248314682576952)
(4694168104486978964, -1664070531819228493668878632)
(1069989532739878064, -8915349329594500128629447032)
(3354706217707065254264, -194303573797498707914736017331832)
(4930772184078574064, -2744160461736608124862607032)
(-731521848874093816, -12915413906143775265186074512)
(5117490435914674664, 3528894980250670819946309768)
(4810672666846125464, 2216216846490160354854150368)
(17817212686720462664, 70054079146741990435484818568)
(3216867450842062864, -2357036498397366645819730232)
(103466385306023966996/289, -52330860709853456898780526860716/4913)
(68979723666771081382526/9409, 11658418300355553491173878852162164/912673)
(923375241833105864, 9288924001108939869090216368)
(6642842996605639724054/961, 326484787968219388420588812416588/29791)
(3386328218200920944, -1652929648328577837336935992)
(-3619889933139968656, 16197684909701806675554283208)
(33364135133016849392, -188742405335756459751036991672)

Commentary

Specialization at T = 135/188 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -951794051889111932592T^8 + 44931125401781237315136T^7 - 935996917001646313472256T^6 + 350884544094515993044032T^5 - 1398456319949084806406280T^4 + 1757156899789831824711432T^3 + 1161615213018494249920164T^2 - 1209850628288081766896304T - 608061871140582830645232, A6 = 16573004223094294578838161979776T^12 - 853911819873454385971944861242112T^11 + 10730626760581365984133349196446976T^10 + 28681501792522102389650863794343680T^9 + 1144251410928840595115561345197638240T^8 - 1598280314283758871148427065729397664T^7 - 426571032276915684981278470103405232T^6 + 1752000471251825933878865739300711456T^5 + 357646791216195149324748670252742580T^4 - 1462595841433547195589341024424494400T^3 - 249356805352454598151173826415660496T^2 + 545782915696199470675908824491330752T + 182530075488483105810112141784098944. It is a 2-neighbour of another fibration built here: 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 2026-10-19 01:29:06 UTC · history