curve #4597

y² + xy + y = x³ + x² − 469161442807105767580908856558571396x + 92434269287007095804145861635730650989170854092919429

a-invariants: [1, 1, 1, -469161442807105767580908856558571396, 92434269287007095804145861635730650989170854092919429]
c-invariants: [22519749254741076843883625114811427009, -79863208663974130943680143863829358783770806297368089377]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 90768017431890829871720181326988326138150445714817351174485656583597333916013927166435420090
discriminant (Δ): 2918121190663854725137527318737780000313483980914697560908953701977099429994158324256020168477002138087731200
Faltings height: 19.5220
naive height: 258.0224
primes of bad reduction: 2, 3, 5, 7, 11, 23, 97, 199, 46457, 16941111922860221119, 112454062659905351733878800067433744181683847668968935740457
regulator: 554782003118680460972586497.1515206298403419871367707513958383538052776830983358032387189
submitted by: apkorte
submitted at: 2026-10-18 20:16:51 UTC
last updated: 2026-10-18 20:16:51 UTC

Witness: 25 independent points log in to add more points →

(723548309199517371, 362997067423143889126928983)
(568405463460022653, 96971598062688468392176657)
(14550727989162127229/25, -16073856801762031302825392827/125)
(19581806917244632989/25, 56665516434998667970310535077/125)
(1413900526547227101, -1501877486978420962974101807)
(198005550919029419, 85444392385297856563679975)
(970837330451231021, -742962583741882220175619055)
(548247988709514789, 2749728956960097116901265)
(-163767047928267907, 406048371002761176018424849)
(203882427224494269, -72494753703438808518337775)
(579222797309129213, -122531480351660387188809743)
(-151509936783396171, -400048671579870400016164295)
(17531741523249803589/16, -60510706762148192633349745265/64)
(-758331324501112947, -110105515181933255019439343)
(1276722072287975729, -1254801600234320017691068735)
(192640181601374461, -95936682516653240935893775)
(14568471002587364911/9, -51046741087510917351674501015/27)
(-753746802413338021, -133543394284417043000614985)
(-392196399182366979, -464877215819358254026615055)
(160742945283591793461/1849, 18183429229091142287929379370115/79507)
(1934072226903783376389/3481, 11763841097931723450460909308275/205379)
(598644311153285697, -161593942939671575747576579)
(671242672796623629, 282759087461665193689028065)
(131632813586869577567693/214369, -18799529238662469867377002941642913/99252847)
(2353162331571404631/4, -1132220008569243924936375205/8)

Commentary

Specialization at T = -197/83 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -131683424850520935576768T^8 - 403203691800547711440768T^7 + 228773565844968202575408T^6 + 1001632750223305381512624T^5 - 240565012803414935795880T^4 + 1113998583596965801461504T^3 - 955469793798674292115872T^2 - 990503756503270017161688T - 226533255431841379251243, A6 = 19076424418751512058704915856577792T^12 + 97086137010965231517170399109938688T^11 + 56104518594343130212702088417622912T^10 - 478767125533383624912178730778518400T^9 - 714366843201118889959457857278267504T^8 + 1042906926398973532541608382597231424T^7 + 1920199847343299876864028118527850608T^6 - 1651848617855242258127597768465390256T^5 - 1294346250171073842969926939033547144T^4 + 242314299085392499926732997139536320T^3 + 623077991032626826626842144461126512T^2 + 269831263748284750614515352678364488T + 41099644923330823113323890449407142. 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 2026-10-18 20:16:51 UTC · history