curve #4179

y² + xy = x³ + x² − 112062306318513926203x + 422716991190152888926575022753

a-invariants: [1, 1, 0, -112062306318513926203, 422716991190152888926575022753]
c-invariants: [5378990703288668457769, -365227480428634526307225833091797]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 284098332150954535104446062634390597697230108860266690
discriminant (Δ): 12871606201980117497887574181788192339090703902217951089160300
Faltings height: 10.4681
naive height: 150.1104
primes of bad reduction: 2, 3, 5, 7, 17, 37, 2150793641842338822806011527249531362686275333941
regulator: 20703650039958719.9109671420581607047404250667051614815127891469886705359
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:59:07 UTC
last updated: 2026-10-11 22:59:07 UTC

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

(34416185164/9, -6043927238420693/27)
(28039679056/9, -8699965315582939/27)
(-57224957478671/20736, -2517725523162991709233/2985984)
(-13206195460096579/1125721, -418618654622388686380769/1194389981)
(-10196057761688693/1868689, 2385039517144374747209951/2554497863)
(-2039706779003/169, -287527163413701056/2197)
(-227151683803274/54289, 11442946414935774453791/12649337)
(82625389, 643007239842163)
(1513573713237259/366025, 38374453957038032279813/221445125)
(5002302935013484/1100401, 98185021278866773395257/1154320649)
(15950588579, 1641158179514993)
(885942921469/16, -819567271166800703/64)
(366496363464886/9, -7016241417218312371819/27)
(8791037975966968/564001, 664721466849191254916335/423564751)
(144234631237316/18769, 318780886489690745539/2571353)
(67335971851334/9025, -37871202821517890063/857375)
(6084748851381/16, -15003676380061137833/64)

Commentary

Mestre construction: sextuple [-1053, -495, 8, 39, 456, 1045], T = 153; y^2 = r(x) with p(x-T)p(x+T) = g(x)^2 - r(x), p(x) = prod(x - a_i), on the locus sum a_i = 0, 12 p5 = 5 p2 p3. Found by a discriminant-first sweep + Nagao-sum rescoring + ratpoints search on the quartic.

last edited by Andrii Prokhorenkov at 2026-10-11 22:59:07 UTC · history