curve #5857

y² + xy = x³ − x² − 470797294546809931786582609675863774717x + 3578632840434933868043848837392154502463568848852009729041

a-invariants: [1, -1, 0, -470797294546809931786582609675863774717, 3578632840434933868043848837392154502463568848852009729041]
c-invariants: [22598270138246876725755965264441461186425, -3091938774135782861888193179884710544862621641718149830552525]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 34323368201782394984208127638439954785918722081299558491309614362603772663830173776217449523450
discriminant (Δ): 1146088071359751368030739931422069120842853922437098210372521071947903485016347662072769386018681723258204178876942500
Faltings height: 21.1981
naive height: 278.7561
primes of bad reduction: 2, 3, 5, 7, 17, 19, 23, 31, 41, 53, 109, 199756568527978213006300391802901492439260471049243347550166556625558371368379241
regulator: 454354993618082092382647406.553508719083677861087849307569268353229909795760670757535337614
submitted by: apkorte
submitted at: 2026-10-25 13:59:37 UTC
last updated: 2026-10-25 13:59:37 UTC

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

(78352946984405527429/16, -2386563282101935396942698543443/64)
(247995563604802454989/16, 144255388648508467040217401107/64)
(-5598617043887373386, 77710750058910290398742487853)
(225921807525821339971/4, 3171500805073444925988969395009/8)
(5855886099467327614, -31976617272311941075707357647)
(23194042516291530934, 71669318222025253263977696893)
(65715281271664477891/4, -133441369428148572550912692271/8)
(6559996047010622749, 27793962559312285662138026623)
(-24690001981579229756, 12316153747525868505278027143)
(-190308526252199683304/9, -1724439486378826130987996454949/27)
(2390571419506839084695/4, 116807254896549807214221375213683/8)
(702897260016237978589/16, 16649884065482710436259987864907/64)
(3541484741833691164, -44223626167969891184740298867)
(-2370548688199125356, 68420460745831254518669033413)
(1421553580225475786665/9, 53113978163939210532228512566937/27)
(-532013573443111292399/64, 42584788498197767667485687875901/512)
(-16145883164490052511, -83492562811965574414164527717)
(132240946444835671285/16, -1016092436171423210353300784891/64)
(26669889491293041659, 99961793564006538923738241748)
(2504129792642553129319, 125305172427854216523073721357668)
(-10860310658656010954216/441, -142736759292749995107226774236947/9261)
(-12108791201003368381, 86625595604295971326393792393)
(44074794974551443229, 261624705778800538244151134308)
(15868856895810388324, 10184464411664673871267720903)
(-12304134874879571891, -86652420183065273617302510257)
(-3046971747248545241, 70603475112053044222089742948)

Commentary

Specialization at T = 261/88 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -52356648678628357469952T^8 - 235519746807875124250368T^7 - 24545191841476895540304T^6 + 811577658055960647489456T^5 + 1079198714145333325459320T^4 + 1254987224811203628397824T^3 - 2522642567144527246899024T^2 - 3309712733210337229896792T - 1026058563214366117251627, A6 = 4825422219471759579683602107703296T^12 + 34580942274254747650686461570052096T^11 + 31234725928512794727685620659061504T^10 - 273042568629290335756659313007525760T^9 - 455151540100143399461517346503626160T^8 + 1043526152788429910684337802802723904T^7 + 1849381307726983593819115896264667824T^6 - 2215248076398208412328290355259090224T^5 - 3463947301549821564345287894913712680T^4 + 637485311836301536434086156313792000T^3 + 3271887185791747167987145751440802304T^2 + 1998129610230740561490595569733249224T + 403271364874736900390724325685972646. It is a 2-neighbour of the fibration given in curve#4647: 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 2026-10-25 13:59:37 UTC · history