curve #4198

y² + xy + y = x³ + x² − 1086310517200696866630x + 13352514607216860921549454188402

a-invariants: [1, 1, 1, -1086310517200696866630, 13352514607216860921549454188402]
c-invariants: [52142904825633449598241, -11536572621026439622410979290766289]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1305893641466358749805
discriminant (Δ): 5021947235420975943571743580817827086198001380263159470424300625
Faltings height: 10.9937 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 7
naive height: 156.9248
primes of bad reduction: 3, 5, 7, 11, 13, 19, 37, 53, 89, 97, 107, 661, 3823
regulator: 83638.78528174318139742384090822535161648673795278030
submitted by: Manvir Jaswal
submitted at: 2026-10-12 08:26:54 UTC
last updated: 2026-10-12 08:26:54 UTC

Witness: 7 independent points log in to add more points →

(-64492440843080/26569, -17309484833994854318173/4330747)
(-7772716885619633/206116, -77769847355469814512031/93576664)
(-592271058307913/15876, -2799099553421777254559/2000376)
(195507568243/9, 1993122037276922/27)
(-70201847276497/14884, -7783034170646904489237/1815848)
(2564602194184837/163216, -26610456661197981786761/65939264)
(46413988937884452/358801, -9701226027493489625115461/214921799)

Commentary

AI-assisted (Claude) search: Mestre-Nagao sieve over the universal Z/2xZ/4 family y^2 = x(x+a^2)(x+b^2) with (a,b) = (488395, 148148), then 2-descent (PARI ellrank, checked with mwrank). Rank exactly 7.

last edited by Manvir Jaswal at 2026-10-12 08:26:55 UTC · history