curve #3193

y² + xy = x³ − 7563016932293571961417207451x + 386369359779887273958016110137382888086481

a-invariants: [1, 0, 0, -7563016932293571961417207451, 386369359779887273958016110137382888086481]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 37047639080308908544920171799389911308078271310
discriminant (Δ): -36803196583049775813483789459555313264990161929395173729711391879394432512725511065600
Faltings height: 15.1174
naive height: 205.0384
primes of bad reduction: 2, 3, 5, 7, 11, 29, 109, 113, 157, 173, 541, 827, 2083, 4231, 5171, 14177, 1708781
regulator: 7284355431267436.97411033272925033869205628705858532887334812345943
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:01 UTC
last updated: 2026-09-26 20:42:01 UTC

Witness: 14 independent points log in to add more points →

(66767376437326, 423140231789716881937)
(-9144873957554, -674364453464665338863)
(12529961180047156/81, 1245305883208100001392323/729)
(-104289251220674, -202076618783126154863)
(-101540199122414, -327716768598151710983)
(-47091677018864, -798807075201939840953)
(17546944193347186/289, 1909121817951638820440069/4913)
(6496704677251840/49, 449100315418851513581233/343)
(-105826098639194, -39625584501587663903)
(256380787837048954/3721, 99553155072107759480970913/226981)
(41025161952201947614/2518569, 2067236229026733299084170388891/3996969003)
(255111346692991103824/16834609, 36237644966030676192257928030959/69072400727)
(39628940233241058646834/267878689, 6939397994014654564678230871020311/4384370502863)
(17246400470544478/67081, 10772216401655188406489710603/17373979)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 72). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:01 UTC · history