curve #6146

y² + xy = x³ − x² − 115631032201418165888424144147067849909x + 459980801783176392424981816718783913220644919243319937049

a-invariants

[1, -1, 0, -115631032201418165888424144147067849909, 459980801783176392424981816718783913220644919243319937049]

c-invariants

[5550289545668071962644358919059256795641, -397423412740664403030207986689522977190737595090461770029965]

rank (lower bound)

≥ 26

torsion subgroup

trivial

conductor (N)

5565696986327983746982011059177077905517699661318627919982786727589293124997375643000259903434698078318

discriminant (Δ)

7543555319902254001698026317800598847732620071939122721750519001436679582173851477539014118207360043671676374941932 ★ record for rank ≥ 26

Faltings height

20.8060 ★ record for rank ≥ 26

naive height

274.5440 ★ record for rank ≥ 26

primes of bad reduction

2, 7, 13, 31, 41, 67, 7877, 1377045863, 33903005927687782367, 139502606190933072261323050259535182425987825909385855854473203

regulator

28964405960541451616748516862.8998304245848320951676963696036399446744908760388074047303026

submitted by

Ritik Jain

submitted at

2026-10-29 13:48:05 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-29 13:56:08 UTC

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

(83441942260394992595434/529, 24048646403764910261771062405813683/12167)
(1523742285184137985557222057773516/170357601556321, 26430984759849112249895978663693472070078575219855/2223526325206874443631)
(-68453051148756322050311/6400, 11137013975781700018383243366123627/512000)
(19310226468472403902138/2209, -1123864537531516849572622604283881/103823)
(-40068821802961762918067/11449, -35118287866268439372998578780389636/1225043)
(-714489336113856742024235/218089, 2887231733457041384978684740894945749/101847563)
(4371560532207671732, -6167303629026398139915395085)
(3170393296074803513572/441, 4659697778630269884287460062485/9261)
(20707025183286845775840/2809, 425663348661804883558180382752609/148877)
(-1164267391711013465, -24352167377037530561041129483)
(-1150568471348960674984210111112602016513759097305164606485587/98858517106561175959645828262803253619984, -470628557703004359937055981634288278266298124590828373756282771248202274765589221382289869/31082871823925964773803636012464819875862409596198898460108352)
(11404433989838076545413/1521, 226005196960181778261347644399345/59319)
(-336976112061273308310713/75076, -613190241494960475469712021363777925/20570824)
(336173276958687773999290/522729, -7424082071101164970982275759998753161/377933067)
(-61047110775413952356/81, -17045200442962746437289241098583/729)
(6124214704877809906236/5041, -6415458116843173739178480669467545/357911)
(1695557681440723032117127478548253981961/340334472308185882921, 17263690601805558521346856923157651845112296307199693633305/6278543552692479357599357986219)
(21649186445109849418, -90018671409352783298382741183)
(29980699977662683581683/6724, -3169879767736148655003766388718111/551368)
(45469270474174345069/9, 58832831051374821826405303453/27)
(20026316547178220884929/64, 2832358919859489106394179471871123/512)
(982558686719133085460677/17424, 957352479146889153727467897469571981/2299968)
(367519907872181890474/25, -5501635843907644345620796031367/125)
(43941041976493630033/64, -9992721659577417636733907310885/512)
(556890910242825890211624/10201, -408037504181282474746185805798843057/1030301)
(490618530353575634706861/9025, 337361030657964255941278074120602156/857375)

Commentary

Specialization at T = -5004/1249 of the elliptic fibration of X1152 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 236727T^2 + 1698216T + 7091904, p = -2349747T^2 - 4768158T + 1680784, q = 1606404T^2 - 1744307T - 4611, D = -430215432894T^4 - 1566242357373T^3 - 3838777968231T^2 - 2986954200284T - 982738454432, E = -1369689721920T^4 - 7511308637400T^3 + 7118753331870T^2 + 4675836198635T - 2198247671215.

last edited by Andrii Prokhorenkov at 2026-10-29 17:16:34 UTC · history