curve #3200

y² + xy = x³ + 296061681592456544047356939873777210x + 5495441923316939043111329873609778827340053867929755492

a-invariants: [1, 0, 0, 296061681592456544047356939873777210, 5495441923316939043111329873609778827340053867929755492]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 23693473551183573088015644406641907435184135279878499727924510
discriminant (Δ): -13048009830206096170282046578924495340327005218624863678705108520539536411898146062737907277503381504000000000000
Faltings height: 20.1919
naive height: 265.6102
primes of bad reduction: 2, 3, 5, 7, 19, 37, 41, 79, 113, 193, 233, 241, 251, 269, 479, 727, 751, 853, 1153, 1291, 1373, 2273, 7283
regulator: 159564718935910448.85798915472698949688920696463176279611406120327048
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:22 UTC
last updated: 2026-09-26 20:42:22 UTC

Witness: 15 independent points log in to add more points →

(-1707441811914673476, 110138399920174438467661338)
(16214483994960393324, 65370005857555056679736680938)
(746315197099198400016/2809, 352082505364800255629069394125826/148877)
(270182225276440575355404/44521, 142710829757887895214822369574564878/9393931)
(655266542151083753126124/22801, 530585124406304125119553064039617638/3442951)
(806131693786103131356/1369, 122763837782268842987429873616114/50653)
(-91964458657777773429684/162409, 148477221703908170356903721021970126/65450827)
(-450086322360134645610516/271441, 94405134443694043691118236118179418/141420761)
(-143200686165823869867396/96721, -40487490236630616737121802218071322/30080231)
(-873226809807839466175628226/649689121, -27056237418925394812010522599894825697388/16559926005169)
(113017516280882017992466524/446096641, -22270064594626083497848273244898091008342/9422007154561)
(108609196040183374863262083217804/11812893386121, -1137846794915182729008350482440717927750337580078/40600784626270629669)
(2366505932682481383669013284/3500641, -115122852213689253049917786882231239216682/6549699311)
(1118526413682583359380556/5583769, -31119840120239262416673972767628958914/13194446147)
(-305168440765452622608957700021743834/178800301712535409, -10236535032977454682528258066515356519201738413121676/75605322622096010606914423)

Commentary

2-isogenous image of D-z2old141516171819-031, rational kernel x=13670258741780008607/4. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:22 UTC · history