curve #3179

y² + xy + y = x³ + x² − 284223355086479201781741517789301698531784400402276823264x + 1795273070993309604429144564052942175172562915080364861669114008722745205559477017601

a-invariants: [1, 1, 1, -284223355086479201781741517789301698531784400402276823264, 1795273070993309604429144564052942175172562915080364861669114008722745205559477017601]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/4ℤ
conductor (N): 15066602282658023728483787194675827223128239968819300284366123156642
discriminant (Δ): 77126710571229910520033250011359173516432798639696348320065892426616832533786908803418997531033598303222554675322897483377998391871825815437341526087177269628308096548864
Faltings height: 31.3745
naive height: 401.5817
primes of bad reduction: 2, 3, 7, 11, 13, 31, 41, 43, 47, 71, 79, 83, 103, 109, 113, 151, 211, 229, 401, 863, 5879, 61519, 169991, 1702817, 893629932007
regulator: 1421518964274469.720327102132152321010586865842530500434343031036
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:07 UTC
last updated: 2026-09-26 20:41:07 UTC

Witness: 13 independent points log in to add more points →

(385496609315749408609376868693, 239123331882085908795731854660301679907025461)
(-18968535870874970339841890147, -601326226579389214419745559072337763363659)
(91451660676881912814205409045, 27214884987075469097642448040693294649594997)
(7275041860595260281583829845, 335524713528058123897295485527725479736437)
(-2419783919884453859414015555, -1571261748819393789303920738760860055108723)
(3910840647682747981579300629, 862285252052882206778025083899962904695925)
(7521014414523279304910639381, 288194809635800604448728101810817284232309)
(784318941707968502084419733, 1254126703561001175476611236447601779569781)
(-18928881084933733084227040123, -626926634791280467694160764880189144137355)
(-5536060941530726459594611723, -1788597733410534924868036128558681020732523)
(12631413381351557752142849813, 469579894789089375474615757410497400751221)
(-627958389623764394436901171, -1404815250281336994539578044768481042947915)
(12673722046261993285751494381, 478329851855161085196678850064794332758709)

Commentary

Elkies - Klagsbrun (2020). Source: https://dujella.github.io/tors/z4.html (curve entry 3). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:08 UTC · history