curve #3233

y² + xy = x³ − 6202714082793801020055426948910463800236372285x + 132953186020261144369696727397524699418485201397577570180837834102641

a-invariants: [1, 0, 0, -6202714082793801020055426948910463800236372285, 132953186020261144369696727397524699418485201397577570180837834102641]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 705477490224703868376944408577475677167161370706714635492150217886150043262
discriminant (Δ): 7636762545126770338989606109037973653861355175864356894137102381626275813656299956606369853773072712966678810501701008807191731896651676672
Faltings height: 25.3554
naive height: 327.9376
primes of bad reduction: 2, 3, 7, 13, 17, 19, 29, 31, 37, 43, 53, 59, 71, 83, 131, 139, 167, 193, 389, 709, 977, 3359, 6221, 23813, 126859, 431099, 7562663, 274307221
regulator: 22523855101581730166738.281528984433324677203667605062150528023256091880291
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:53 UTC
last updated: 2026-09-26 20:43:53 UTC

Witness: 18 independent points log in to add more points →

(188816805689963246679138, -75454840115971021778686393251951057)
(65760811771178383036278, -3072340325017015594278819368014461)
(-35025057070079700221094, -17528161628808998148544372091802369)
(-930169119586479703533444/25, -2208750964473311797342546648514754861/125)
(152442484279096528649154/49, 3657194176695897172225121183156672313/343)
(1111768532699863040041402/9, -955262661118875793523357353614684859/27)
(190215124338605208938696526/11881, 7956925713182797409661410471848708800327/1295029)
(-30111812217667130780279046/1369, -814779006826608417887645487903465656061/50653)
(10929233723472262976171994/529, -44907051218082977551330141908299588679/12167)
(205669165508755730883686562/2209, 1976829126586539674720955183044614267329/103823)
(3225024458573384862628756313804790630/23134281855721, -4948022765854651924803737341783639565931515697268660191/111271523346747278731)
(1864160019540380534917556394/9409, -74600428899021046307745058241620247100801/912673)
(-2423123755763986637356505382/44521, -165215864722450228021852312939956319807851/9393931)
(-308597698857739214034492712619466571635084542049/5181746050134429674321284, -201259153433387092447530975030606886619216563739223220930303546916328237/11795441133288048325066603090695517048)
(68755055810882396807727368562/797449, 11011093717574346093749793099639380518943883/712121957)
(19167635610634759819828682307826976966451757497445673568297555306/24586069869248464595744617126476086576641, -2640504675668379299208805822718501043428429325140891213662562346380399994159074340542858272824969/3855082417316462021833826615079434650144472721118104274274561)
(1723424800702763480729462100637225100512031796364724834/25372549743526700788133078582281, -639340443829027682891070711646108331471965903940166160354410500124932826035592395/127804506864508756000315893303999785983314604571)
(15348719971848156100916059244313887242194391734655289842059121805252459938815918/123789715464175453525830440996232531046500954813520167529, -49083736721911808936315092010219481548265151351781744152865872678169600900305623083829973859181725287091257504731042627/1377294607307668501910232454100326599023186274564095207990714066813262677418824398667)

Commentary

2-isogenous image of D-z2old141516171819-008, rational kernel x=43921119967229347350882. 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 20:43:53 UTC · history