curve #3215

y² + xy = x³ − 26203137369313475904909479991597350x + 1632588248343784552178712665324524152998828231119332

a-invariants: [1, 0, 0, -26203137369313475904909479991597350, 1632588248343784552178712665324524152998828231119332]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 31049725562981000812793039146883386111013163212447670
discriminant (Δ): 7359860355460368294091468531759949455770077126048095552372455462729829854065651076082693193219520000
Faltings height: 18.3786
naive height: 249.3672
primes of bad reduction: 2, 3, 5, 31, 43, 53, 67, 83, 97, 113, 227, 271, 397, 449, 569, 881, 929, 2909, 4877, 29347
regulator: 7820926100431308.9977356513503967074995643133582821923459370835064528
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:03 UTC
last updated: 2026-09-26 20:43:03 UTC

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

(59549633938477024, 16833700528201574270280118)
(142046582091770104, 27868062533359288629172798)
(93274704391961212, 64696982221012348829530)
(78145765592565948, 7882988374248474895119786)
(684958047878659231/4, 372393506811855101207034689/8)
(67681777904086624, 13005828409820242938903658)
(18545044388765704, 33956264357289240390706798)
(93878451789280234, 210818795047112930964238)
(-92860659540334376, -57140895283433808760604342)
(44759254234233748, 23439830203467504321316906)
(93314292004238524, 23697769720367464594858)
(109253370530755312, 8595816756811177577059558)
(1463036831810415877/16, 67986161922212409371476357/64)
(11355408569171190402484/96721, 397347003746862898704297622297838/30080231)
(53034961459735260724/1521, 1636729460945517094620676211342/59319)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 49). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:03 UTC · history