curve #2728

y² + xy = x³ − 333795891475292713173950542822194359898759421775x + 74228313544231762777712368957403490860668082434004048855306579752017657

a-invariants: [1, 0, 0, -333795891475292713173950542822194359898759421775, 74228313544231762777712368957403490860668082434004048855306579752017657]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 2987967270845792978421324415770
discriminant (Δ): 3514893475251583789589985767621001993912881097478087443547037308249099299962733036396071139758796633726257347707217215545988769382400000
Faltings height: 25.7433
naive height: 339.8942
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 37, 41, 79, 107, 1613, 1801, 97511, 1611289
regulator: 28636914.29118202423707511267144661765735412193
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:40 UTC
last updated: 2026-09-26 20:19:40 UTC

Witness: 4 independent points log in to add more points →

(247917146828997382687143720444917083275119406289/742826828116078394008576, -117570329842267135422911976182807887319393955923946113738324201410103/640223145272908184070179587060301824)
(446889245784534245946136559785160475141274575167509066/1336002477617657319753548879329, 1441557911710121062063359576046861661990963263083419833779722878757907179272309/1544226124275244127092395420228982799256689233)
(114863167822441289348441644041707270394132480219163096348336489100831634090406/199064967457659882636721511144799522865260650041420369, 24117662339303375427363867136180035455898358846211606204786518097260853497750741746680162394661463662611182267718251/88816213885453375466442031170047269745155032011278792594457882959414963131319847)
(12781861363362559750597491812574986528710485769961974/39904287059765722302765462025, 104966716700929713259791461650878790419250363493745704135266874172559068940303/7971303301599041640078592646751861594734875)

Commentary

2-isogenous image of D-z10-024, rational kernel x=-2668575461961110890446385/4. Source: https://dujella.github.io/tors/z10.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 20:19:40 UTC · history