curve #4176

y² + xy = x³ − x² − 36028631803163826514x + 79239789780197674490101502104

a-invariants: [1, -1, 0, -36028631803163826514, 79239789780197674490101502104]
c-invariants: [1729374326551863672681, -68463178362308606289964311290805]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 972088439459265420913132639723426333713908127782
discriminant (Δ): 280610246419042224963014060069946123138051782663560804584172
Faltings height: 10.1630
naive height: 146.7061
primes of bad reduction: 2, 11, 17, 23, 73, 79, 1781409251921861107956807775400919860443
regulator: 110035839702498018224.139710525457658310613966769372933811140405639022206
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:58:52 UTC
last updated: 2026-10-11 22:58:52 UTC

Witness: 17 independent points log in to add more points →

(-25760533661/4, 1681213937591437/8)
(4384960201, 74624505596909)
(-960189547044928983/369062521, 2794653025188616076922382390/7090060090931)
(-45768804057583279/7946761, 6930119208599976766754190/22401919259)
(-367753117611029/79524, 8596952608472635850869/22425768)
(-2728127374946/18225, 715752049081320220022/2460375)
(366180957168222/78961, 2419637922487750002440/22188041)
(1943184730049913/273529, 60999490549325815574566/143055667)
(2230632657713271/160801, 96697205856269504025424/64481201)
(4292047034624154/3667225, 1381098127278546717459412/7022735875)
(4465156631265598/168921, 291231187658335145356072/69426531)
(21434398101410302/9504889, 2850113796407177453827240/29303572787)
(77863997123224733/15342889, 9893452720016847250774190/60098096213)
(273378464511883876/112225, 142937108834364196487337744/37595375)
(1723405473484986036/402764761, 472681799195052644061543860/8083085988509)
(2357180103898667556/61701025, 3576650327240990533465091956/484661551375)
(168629120102921127604/14185524609, 1949023322590700486259844622188/1689538537505727)

Commentary

Specialization at T = 3 of the K3 elliptic fibration y^2 - (Lx+B)y = x(x+D)(x+E) given in the commentary of database curve #3589; points from PARI ellrank.

last edited by Andrii Prokhorenkov at 2026-10-11 22:58:52 UTC · history