curve #3548

y² + xy = x³ − 100111108435616174555x + 826209068409136115657230713681

a-invariants: [1, 0, 0, -100111108435616174555, 826209068409136115657230713681]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 589032177631665151529373868401860082 ☆ record for ℤ/2ℤ torsion, rank ≥ 13
discriminant (Δ): -230678890175850163494897373947822006554841501765549295673415168
Faltings height: 10.6555
naive height: 151.2964
primes of bad reduction: 2, 3, 17, 59, 157, 257, 307, 593, 619, 827, 1543, 1723, 3779
regulator: 7924209813646122.725333010959841880320666942846401699418973847465
submitted by: Steps Unbounded
submitted at: 2026-09-28 05:53:54 UTC
last updated: 2026-09-28 05:53:54 UTC

Witness: 13 independent points log in to add more points →

(-6923545130, 1089704983380481)
(-75734573660904290/8743849, 26412373832897842852693877/25855561493)
(-37987561673914250/3644281, 5972975019530810857743661/6956932429)
(-209392122207626/18769, 1914832759591695265793/2571353)
(-27170162822276/3721, 245297134894381095205/226981)
(324396657982/121, 1011106619316251459/1331)
(27011913353783934/444889, 4387084264249059241457523/296740963)
(310744747367841730/380689, 173210077164481318041739037/234885113)
(9566981515046105323/679853476, 26316774319426753021958885671/17726499533224)
(5138353743499418566/632975281, 11793890983774828908041452639/15925025094679)
(9012035800673747554/265722601, 26148097401636145936899844025/4331544118901)
(34527501970803266303734/4392197403049, 6669713424560858920876488193581389/9204978452821763093)
(3905840612654955316657492/355350813264001, 6876282893271163078768337445419214629/6698629698487180114751)

Commentary

2-isogenous partner of curve#3547 (Fermigier 2-torsion construction; Dujella, Proc. Japan Acad. 78 (2002)). Rank exactly 13 by 2-descent: 13 independent points, 2-Selmer upper bound 13. Torsion Z/2Z, root number -1.

last edited by Steps Unbounded at 2026-09-28 05:57:23 UTC · history