curve #2730

y² + xy = x³ − 9548381337509275543430232516717777241248252565x + 377663828681605388165662032646784651619190173701890369766128592660321

a-invariants: [1, 0, 0, -9548381337509275543430232516717777241248252565, 377663828681605388165662032646784651619190173701890369766128592660321]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 763384827925528186459799976258
discriminant (Δ): -5901517358351803348368744952578871502995381497346613443643396550315951162795247782767649156689773294385517057042862890306464137726119641088
Faltings height: 25.3823
naive height: 329.3324
primes of bad reduction: 2, 3, 7, 11, 13, 17, 23, 37, 41, 59, 79, 101, 131, 439, 1901, 4163681
regulator: 1498194.032403684042537374664115676953141968527
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:43 UTC
last updated: 2026-09-26 20:19:43 UTC

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

(9533146388344013080424422882/227529, 776228258355276017285491239695684479112659/108531333)
(27510916627425275007607725344914/396209025, -55450345866065844698197155751876257600336498697/7886540642625)
(45428941003353162664779475629185959471389196637315618362584526971842/8459767238371502648537753352041554715218009, 306145257621983760049880172345317285114975391963653917190468176883288869531968564145904431681551097093/24605807444758116078074103437873769134330331519840918006158981027)
(-100118220993960898126186180741820232840981729205246/931625709524483383444725625, -362671469095748395211471276382255202583678771423801204872226165676600812683/28435586974398536014008539902294651421875)

Commentary

2-isogenous image of D-z10-018, rational kernel x=907799169750913420899183/4. 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 20:19:43 UTC · history