curve #2945

y² + xy = x³ + 480419394415878792244194487343807452105x + 11715225724504561983812913629070384220968219735916971578025

a-invariants: [1, 0, 0, 480419394415878792244194487343807452105, 11715225724504561983812913629070384220968219735916971578025]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1041840789186569790031551162756230429730
discriminant (Δ): -66386950833066730499892908623042282494252906337179088693609537500582290865032835849655240215224426941940274197607936000
Faltings height: 21.4819
naive height: 280.9396
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 61, 67, 107, 151, 167, 199, 587, 1181, 613667
regulator: 316058352.724029503011820555727648868697696765573285054
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:31 UTC
last updated: 2026-09-26 20:29:31 UTC

Witness: 8 independent points log in to add more points →

(12564125617806673607830/169, 1487373900716243593804898382690025/2197)
(39705014395620169110144682/1692601, 417194478284574162006261349439691156281/2202073901)
(-100734366653147000414550/11881, -108600645449747251423478525259433155/1295029)
(-1522723274112636721467290/657721, 54893674053262778714178969444806823535/533411731)
(3258276462809291489878/49, -199318132348444329750369538802909/343)
(-1051881197304195519918890/579121, -45877638313132337367542444157250866595/440711081)
(-941909350291879569087378709364314365241002410/59132510384268358546407889, -2092288103915975204450210742549518897182311232550307145124840308435/454715203410890756201500198868967378713)
(8968061481997083931092388150/196249, 849274169762404314069142119694379670580415/86938307)

Commentary

2-isogenous image of D-z6old678-035, rational kernel x=127566650300506323791/4. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:29:31 UTC · history