curve #3120

y² + xy + y = x³ − x² − 65307967645295697830427336722134356346912165902774467x + 6776556666061653734253097820671526771886795499021732626074384890594739542801859

a-invariants: [1, -1, 1, -65307967645295697830427336722134356346912165902774467, 6776556666061653734253097820671526771886795499021732626074384890594739542801859]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 2603348955675569045873307484526694064917261974747900874693810
discriminant (Δ): -2011174270050636328301112822431930831743045343284277992236255883348383349218526221894927781102384865693501720077763078300447429091225685218518855424000000000000
Faltings height: 29.3202
naive height: 376.5534
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 37, 67, 71, 83, 97, 127, 149, 191, 193, 683, 947, 1481, 1951, 3457, 14008947836626425239
regulator: 2592486636509.80077522205220039024725851696751875887882930881
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:38:13 UTC
last updated: 2026-09-26 20:38:13 UTC

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

(1788826751214181251515755973484767667/1556381401, 2338082497983241577784513418441718023396199558840014966/61400802650851)
(-5312300021776083430523368917/49, -1216688688027523408611134039453543851548962/343)
(-12863776709018956287677733, -2759444712834869445029809501793336961134)
(316843215496912418503345964732967/39601, -5636997539277726057243000021433979315805033489266/7880599)
(832441435507833836100408800572093/1159929, -22677708471845722766044238616688291506982031415172/1249243533)
(282432251269388148216746667, -3295531753207275854427540794423250665534)
(40998841858375946773770894667/81, -7412765162229735171573623893194825156052286/729)
(-144287436906116137982890983, -3632598978213189037122671847288931741634)
(-5020971851185755453273846459702611493/63504504001, -54141603227670898719906951183559579721012222970463799534/16003198512756001)
(-549754073815107967161679644699/100489, -85081388450298598768448014709957725683914804314/31855013)
(4748212234396034210903864020777/19881, -6142853430788805467802701483910296551162143914/2803221)

Commentary

2-isogenous image of D-z2z2old1011121314-046, rational kernel x=2374994013564696008458221867/4. 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 20:38:13 UTC · history