curve #4280

y² + xy = x³ − 838263670694870563703569572806971280x + 295648809904566633705826741132887998607563449009873152

a-invariants: [1, 0, 0, -838263670694870563703569572806971280, 295648809904566633705826741132887998607563449009873152]
c-invariants: [40236656193353787057771339494734621441, -255440571757545571582189288628845911383591829186632335489]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 1087082323124172970804700960653831711732174870850503116296359589462243941425852611957958270
discriminant (Δ): -62038182325295515245525043587379882577889341723004194550221798927482954679027231232148652381616223680000000
Faltings height: 19.4134
naive height: 259.7652
primes of bad reduction: 2, 3, 5, 7, 13, 29, 103, 32341, 188159, 701629, 6336451867, 23295658112341, 3186684232608907, 66377124816574825306552483363
regulator: 11875510971711546190803686.947822306128798340342150300358129417513378322337440097354284
submitted by: apkorte
submitted at: 2026-10-16 14:16:27 UTC
last updated: 2026-10-16 14:16:27 UTC

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

(450135901719031144, -97590070664492191315157072)
(1183673181395391394, 980735442560534736401744428)
(-514905077886240776, 768608918229306950214770608)
(1556095110685815694, -1661086782501013435126951172)
(540099366249334144, 21309016686172532317554928)
(3293193538819169596/9, -5257123631769833483690589044/27)
(-132681059029128896, -636030478105372687241975312)
(93408716061834766912, 902735080177903126755969752560)
(529912990908061384, -15674516904000033423660152)
(650841307475068144, 160514561280104059501026928)
(565257377594983144, 49222017778736174869869928)
(-21386755706022356, -559970327852991671003214572)
(483530372008775644, 58077964457200034526899428)
(8960503153219272829/16, 2745577936015554461734604617/64)
(431635259968694944, -119340626644037679595214672)
(355327548202166944, -206527146414712157254478672)
(531931833869906254, -16142359096224404238001412)
(138256143025808370979804/143641, 33581489416804097870619948526986892/54439939)
(-923135413436773232, -531792331602456759458127104)
(414383205933767344, 139433098291239415527676528)
(546675497184576064, -27691411099041031947812912)
(331641571795101184, -232640715252552249336000752)
(516017073359089984, 22185083773644189254845648)
(520135533447116944, -18869599335684926993238272)

Commentary

Specialization at T = -709/119 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -70781753951590292657527T^8 - 367565467671631265368148T^7 - 564938152357888095468709T^6 + 861671225097878262924926T^5 + 883290795135789128496595T^4 + 1492235942354111474368304T^3 - 390179707279847183534299T^2 - 1933866828077577012667322T - 753991179760473695494012, A6 = 7240259042863556427532625579222554T^12 + 56530289701184340202303695554449444T^11 + 163263464615793597915652884274103291T^10 + 33336026591601761367769399981188740T^9 - 608655863176039416519152300146091520T^8 - 922975179177849633814913734492373280T^7 + 1471644379398400596650029609789831050T^6 + 1259508950348326801233446624368517040T^5 - 1608259867492513771432473315149086770T^4 - 1344292356731452517237148523695223300T^3 + 750512254493731351508294582787338435T^2 + 986666108581324895487891684114883660T + 253245346808298629177743261916441680. 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 2026-10-16 14:16:27 UTC · history