curve #3218

y² + xy = x³ − 726819292649862569939385290345x + 375539705698123987190329314594130495765391961

a-invariants: [1, 0, 0, -726819292649862569939385290345, 375539705698123987190329314594130495765391961]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 23989560829689213420672070983398343053293012701577866
discriminant (Δ): -36351926352773507551734212592902302889800277268922166282069900832990856517581807029476917248
Faltings height: 16.2663
naive height: 218.7970
primes of bad reduction: 2, 3, 7, 71, 79, 89, 103, 257, 821, 1327, 1579, 2543, 2719, 4831, 8369, 15511, 395849
regulator: 46098410676444245.459706886971035453216394189145722584287405840988474
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:10 UTC
last updated: 2026-09-26 20:43:10 UTC

Witness: 15 independent points log in to add more points →

(340993186755166, 12936333282110130727291)
(568969835333086, 12090994900661403086971)
(-25528055317834646/25, -909983836032576135299977/125)
(203753315272042378/361, 82625797267972502329020001/6859)
(1916212243211842522/49, 2651939140060244947288050949/343)
(12501548857791549706/3721, 42981282428133259818912129583/226981)
(3795011245420226878/14161, 23832098742442348769119853117/1685159)
(-33240926223581230044046214/82282348801, -579698603328790360025992007568841169021/23602609471218049)
(366384312483871185022/452929, 5426517925901910668775800114011/304821217)
(151722099166111896432338650364024717884906/274277967763893427427823961, -54272454714773765227847570155380617055992830111964694897479063/4542410555830416092113555251209298919459)
(102580991259600251797654299068813938234/8051669891037163577761, -1036729416034911302254766799365925580176920696902967724589/722485177487822351397247730272241)
(176545340569128577392842730556410/133001422610347201, -2028868528325177685434119764744733739136774277149/48504767749890340592960801)
(-362673372359872285842822632/1442523504601, -40349377693173512257148741039608194889151/1732544297724535651)
(-45329188976457388432207872293272358/648721213331977834801, -341023420721433658100682463442073589964440733664501371/16522933293849658657377131860951)
(177916833418159953360212124634/18466142700625, -74767102954979831113879863662775478895829173/79353170066693265625)

Commentary

2-isogenous image of D-z2old141516171819-048, rational kernel x=8340799720431663/4. 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 20:43:10 UTC · history