curve #4282

y² + xy = x³ + x² − 32614494317959803782x + 57512498176189722855223530564

a-invariants: [1, 1, 0, -32614494317959803782, 57512498176189722855223530564]
c-invariants: [1565495727262070581561, -49690798435969138501378659768941]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 246900641867205563351711407820506530
discriminant (Δ): 791380366031975076649049026230168075911784068575688756250000
Faltings height: 10.2087
naive height: 146.4075
primes of bad reduction: 2, 3, 5, 11, 19, 67, 131, 149, 157, 163, 239, 311, 677, 821, 1051, 1489, 18199
regulator: 126612352065026454.77418916507892316838525967692426429558041204
submitted by: The Wren
submitted at: 2026-10-16 15:26:18 UTC
last updated: 2026-10-16 15:26:18 UTC

Witness: 12 independent points log in to add more points →

(6824944803877/49, -17815112297972151532/343)
(102680490950173/22801, -152674329874979721799/3442951)
(9427998878913/256, 28615328558411259321/4096)
(3601919702213330893/4532641, 6835804068672501958497290964/9649992689)
(14199342146748030558147853/1485052660577161, -45054995139190774860492362944073790309/57228555133065156194341)
(280150903961720392/277729, -148279567376604672138795942/146363183)
(39890853157/9, 146617741568677/27)
(39202476797609431261/8844462025, -4746598959134333858165055301/831777431141125)
(3341693078464345926913/639385747456, -88284397931289521395517963234879/511263073837776896)
(119655895966589184913290472/10277292682182289, 1167646927897110363184898121837533548362/1041880925527618314854087)
(2347900497477270472197/132957224689, 108279873334224410741361639547212/48480591710024137)
(207444899079835308008356/46506739572225, 8873147077674243162779119781888366/317156663585652026625)

Commentary

Obtained from my curve#4259 by a rational 2-isogeny while searching for smaller heights and discriminants. The twelve rational points were transported through the isogeny to this minimal model. PARI/GP returned rank bounds [12,12].

last edited by The Wren at 2026-10-16 15:26:18 UTC · history