curve #3133

y² + xy + y = x³ − x² − 6733405851080577415475454221932585499304047x + 4800325455798548390824144090920449266363788590480449836501600119

a-invariants: [1, -1, 1, -6733405851080577415475454221932585499304047, 4800325455798548390824144090920449266363788590480449836501600119]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 2329609314357493409271842921820896872147183012290
discriminant (Δ): 9583561190810340949965506160233081904365761537906361508834070611789094135130032823242817457223775245723594946978512896000000000000
Faltings height: 23.6479
naive height: 307.4606
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 61, 71, 79, 199, 601, 619, 859, 1123, 7207, 191413, 3560785447
regulator: 118334321463.573146402840827289155616408227237554961878397221
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:50 UTC
last updated: 2026-09-26 20:38:50 UTC

Witness: 11 independent points log in to add more points →

(50429106765918034135627/9, 10209853371013635306144099631896706/27)
(1375362162769333581779723763/543169, 25271915733559758246347818833627891635558/400315553)
(17753007070797729895507475567563/4790439369, 58135309193657036492499090136121538549011430142/331560680046597)
(446496888543804182065307691/570025, 1112404713531836154703264257824331365474/430368875)
(4268560138861462087827, -232021827991116929702294289753914)
(2733669554261469467827, -82595355076513411474712481933914)
(4112932438270069308147, -216059075917969934152568499590234)
(-624081786012809503053, 93591961365784987234330897324966)
(496637464908185004906751323/11449, -11048252638240567810184553779847367808302/1225043)
(15119404109271018481107, -1832821949315141747883603089429114)
(150483958258821153253360347/32761, -1577667833005258823329326427476197416274/5929741)

Commentary

2-isogenous image of D-z2z2old1011121314-044, rational kernel x=-16895950390905374942613/4. Source: https://dujella.github.io/tors/z2z2old1011121314.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 20:38:50 UTC · history