curve #2735

y² + xy = x³ − 756424249574203084341607995076958937142147570x + 8007487662620571545128124458284733393694369249815919711100358400900

a-invariants: [1, 0, 0, -756424249574203084341607995076958937142147570, 8007487662620571545128124458284733393694369249815919711100358400900]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 38426795181551951735578893330
discriminant (Δ): 113394414868097028545430104232888471019043402517946342102171991417040243069502924545107174957564941536406657016310000000000
Faltings height: 24.0315
naive height: 321.6251
primes of bad reduction: 2, 3, 5, 7, 13, 23, 29, 31, 167, 1901, 2731, 5059, 8849, 17539
regulator: 22604962.97691027870480936985998159543062322482
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:58 UTC
last updated: 2026-09-26 20:19:58 UTC

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

(120405351046846516442350733716/7177041, 3801695439501119667737511212636573382099058/19227292839)
(1082751591551438153893561693526570501540/68182385884940041, -4907859094580375363293923134948553622996695098540720470/17803618987260228986560411)
(1983402805614314202111095830784385831442697607248029642640/806439917605326623729390129373241, 88326159131948746305294535180291134767341247307389308912932225028035187396350382913030/22901188680393683605372341483660123693422947892461)
(1252129434600921299137047473681299156972730441144363433999848532262293559390/78855796411026467657519370410127601859112470712578169, 1147106832272269461872509720458516441812756602107963136629289017116616742466980966887831278535976500592509770/22143712460718370738721438819924346666159423214312717284443374518215090570773197)

Commentary

2-isogenous image of D-z10-015, rational kernel x=-7939480428474519917692. 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 20:19:58 UTC · history