curve #4596

y² + xy = x³ − 31180847278435908893557890743856374440x + 66736605343918582336580668895301473212364323476878081600

a-invariants: [1, 0, 0, -31180847278435908893557890743856374440, 66736605343918582336580668895301473212364323476878081600]
c-invariants: [1496680669364923626890778755705105973121, -57660427017145655141050718929587858295818943617580321462081]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 123226310933262615847910201689318930893780182413475658859257710637903282103668438810
discriminant (Δ): 16156955233106339020067315994684713363069757263676035115482748359738116426961633595817413603509684022865920000000
Faltings height: 20.3795
naive height: 270.6122
primes of bad reduction: 2, 3, 5, 7, 13, 23, 29, 31, 103, 191, 5944579, 482763421193147, 443725827229977059, 12448426897419592085712173594953
regulator: 995779081930479609016831.16429224775633921318579735249115984015816824351633912867575679
submitted by: apkorte
submitted at: 2026-10-18 19:42:30 UTC
last updated: 2026-10-18 19:42:30 UTC

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

(3581955085836954320, 1003063007300751780012869720)
(1354052840227682480, -5196026465168235499237183240)
(3023548846909310420, 317196103109692472714875460)
(2901387554057128919120/1369, 161557478196782653684480535644280/50653)
(3670008105755379680, 1316751157377517845252124760)
(9648665918044266320, 25770962419851708948656015960)
(3019731385692404240, -339236244626293408095331240)
(1635697500429606380, 4484472139547938986490185260)
(5228502060843001280, -6829367788294326862944931240)
(1189500813061547480, 5597320907471521973602591760)
(4285614827573920880, -3437909957040905075037847240)
(2523333785192695682, 2030646354852065017200367946)
(179444965314695836880, -2402641942294284442160652729640)
(680263062223363190, -6770540876272920750564911740)
(32608347059082244580, -183637077677278754936078801740)
(-2786633639347924054, 11488563084883270982089360802)
(13921532190997297295/4, -4888668890258850304795007795/8)
(1154491271186673680, 5681316143902100716225641560)
(4507235854782861680, 4214553104836072239231304760)
(-56129581766264455280/9, -116518917552669459151109239480/27)
(3423650262496814480, -337933311967610065077091240)
(2978127463004345600, 538334887088042838477071960)
(969306742077490520, 6117476390343192307083109520)
(3027928025162027600, 290417379293591183930831960)

Commentary

Specialization at T = -89/129 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 19:42:31 UTC · history