curve #4120

y² + xy + y = x³ − x² − 282822262647497336219014806401252894409458843464127030x + 57888571992079360840369014591533726727536212561741309784320032785296140799062597

a-invariants: [1, -1, 1, -282822262647497336219014806401252894409458843464127030, 57888571992079360840369014591533726727536212561741309784320032785296140799062597]
c-invariants: [13575468607079872138512710707260138931654024486278097425, -50015726201156567766078828545995531160731863030037293470981883134052755462138645625]
rank (lower bound): ≥ 21
torsion subgroup: ℤ/2ℤ
conductor (N): 4178638861833190752303116809257681047270030756539116341979925537516195581325565936225850 ☆ record for ℤ/2ℤ torsion, rank ≥ 21
discriminant (Δ): 171117769459803743068767682072176958945198451364160753183162248861211276528863305664316066926965040004276624407303670629457651732009717871980905630188544000000 ☆ record for ℤ/2ℤ torsion, rank ≥ 21
Faltings height: 29.4164 ☆ record for ℤ/2ℤ torsion, rank ≥ 21
naive height: 380.8436 ☆ record for ℤ/2ℤ torsion, rank ≥ 21
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23, 29, 41, 43, 131, 151, 167, 179, 197, 251, 331, 389, 463, 607, 797, 977, 1051, 1427, 4969, 6971, 19381, 125921, 1644949, 6797369, 178110811
regulator: 24287962520379094105944.112291746692964269703295948714549038623209441633673172566
submitted by: wgxli
submitted at: 2026-10-09 20:19:28 UTC
last updated: 2026-10-09 20:19:28 UTC

Witness: 21 independent points log in to add more points →

(305092865918334107454465609, 8172113474855314202219270345010421995)
(305106585588444211135380779, 4228055748249897189827994887555320785)
(303861627377953936126567529, 76530595040845811649804532561687429035)
(305073264264203267049648569, 11715628332957865806581781897334704315)
(309660362509698950259809129, 54012647416162366760378682450547141035)
(310014802095510050795887769, 68933715370407918203008733739182517915)
(310104231965511637492616069, 72468142102786333665919248702036849315)
(310564303496616286133469129, 89766392031748239784619320542493109835)
(310789885056523026935313209, 97867722828421813356419777364323431995)
(284694500910977895316064169, 667376474708739275260022177231933519915)
(287027441448606371475897053, 597918206852086232919671472390085868271)
(290284954395617945455436993, 500488798749770940013810920233388736635)
(290953306677543007621625129, 480427168589683838266474959032489585835)
(291420118704163964497780529, 466398985353220720121970472225900128435)
(295428313594282477335105209, 345293059452288172173495879155348196795)
(318150655106356207703692269, 334128791578651676580690950255231415315)
(330417672192263347396913129, 716047908717779648106802699565540393835)
(339000096846755832758320419, 984911139822489366391908086863128198665)
(246011396771319350248524585, 1788875981458828691226465015966577392171)
(-281057372022911784527489971, 10732018433151036596242108450629438461535)
(7616510087280599447617936329, 663134117107516822433636510698063595317835)

Commentary

specialization at T = -69701/5921 of the rank-9 fibration of X1152 given by: y^2 + (8764T^2 - 671960T + 487701)xy = x(x + (283T - 1919)(27T - 83)(T - 32)(8709T + 279467))(x + 529(69T - 338)(3T + 38)(17T + 149)(87T + 884)).

last edited by wgxli at 2026-10-09 20:19:28 UTC · history