curve #4059

y² + xy = x³ − 325146974748370591977814524288402566846087929390137245x + 49107828061327010750772753090933959312108158966707898552396338432963807516121137

a-invariants: [1, 0, 0, -325146974748370591977814524288402566846087929390137245, 49107828061327010750772753090933959312108158966707898552396338432963807516121137]
c-invariants: [15607054787921788414935097165843323208612220610726587761, -42429163444986537288667658693977523027544131969638270098035421218999060610018544009]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 10440871783541742304283973928724932391253078912604073640431645403950958428574441994981529610
discriminant (Δ): 1158179964417361775568351695694005473217098903122994566856378459503457310809857332089075863659209449861520528072033365424182231547165303577500705206329926656000000
Faltings height: 29.8021
naive height: 381.2620
primes of bad reduction: 2, 3, 5, 11, 31, 37, 47, 67, 71, 79, 83, 137, 269, 271, 461, 659, 797, 1069, 1447, 2389, 2861, 4049, 5851, 14629, 27749, 45503, 522881, 325634333, 12647136419
regulator: 66662257918157868589023.1508276224478262659618000762461513995118445217507979058
submitted by: wgxli
submitted at: 2026-10-08 21:08:37 UTC
last updated: 2026-10-08 21:08:37 UTC

Witness: 20 independent points log in to add more points →

(469685477099134524514766974, 75638783335376039285665865782312354123)
(19342306544446956631351834, 6544155563611415401358742592015763422663)
(29078738974960229152816074, 6299012006055222965965203825142757271423)
(41634347399201492580390562, 5970152068652166535711716304692939710575)
(471290049878053680524873314, 741322384690295147270292487846020408943)
(74047964538363064190645794, 5043547251457287303738334114302373228783)
(105038339016975996370588354, 4014202713294161766743234442238751388463)
(114355609503065120359214704, 3663454569508232060827453065842588185513)
(151290960276188970319760578, 1838184417073788898722841465853497363471)
(485891415359814220201060714, 2415788839552817528666225030595925886143)
(514938329856997973892505264, 4268371986018766118672390548293735273193)
(553036894162491255406648714, 6199659826088295838745600800477663073143)
(570353453619751711486288594, 7014046834480309918149329663556852073183)
(579159436545254061624923578, 7420297940842986063196341373100813163871)
(-218757708338447318455390796, 10477003351458663449845318915098821255533)
(1129149624462514324957450954, 33490448800850053950473084827180123129783)
(5954595012871191586358140714, 457434439114247609672429727863073512021143)
(17834893719053749033027879954, 2380595384506970723972787691683117586751263)
(-1716363932695580699444978207054/2809, 660032788961308325104147466463113715659041091/148877)
(9193782448818600488288801962866/18769, 6981168060591019399786974874913545687455059079/2571353)

Commentary

specialization at T = -539905/1824 of the rank-9 fibration of X440 given by: y^2 + 15(T^2 + 128T - 16)xy = x(x + (127T - 55)(13T - 55)(52T - 75)(7 - T))(x + (178T - 115)(43T - 55)(4T - 3)(8T + 115)).

last edited by wgxli at 2026-10-08 21:08:37 UTC · history