curve #6215

y² + y = x³ − x² − 19492779274097726630x + 33125219238028595154211007592

a-invariants: [0, -1, 1, -19492779274097726630, 33125219238028595154211007592]
c-invariants: [935653405156690878256, -28620189416042785782298165290200]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/5ℤ
conductor (N): 475644963948942997
discriminant (Δ): 28093925619469856865522038767640061031447253797 ☆ record for ℤ/5ℤ torsion, rank ≥ 6
Faltings height: 9.3354
naive height: 144.8633
primes of bad reduction: 11, 89, 109, 281, 701, 1607, 14081
regulator: 2280197.3306330398676630070849603073224781269532635
submitted by: Lewis Combes
submitted at: 2026-10-30 20:58:59 UTC
last updated: 2026-10-30 20:58:59 UTC

Witness: 6 independent points log in to add more points →

(7500728655083176249556/2942573144449, -74610654858821821354038497/5047669373975803457)
(2549140434, 8985631295)
(25268263767, -3959038747959973)
(-1067837158784/225, -462009527047808936/3375)
(208266674005557878625/81288852544, -26409067547391971214427287921/23176427326524928)
(1034664909609337916109/405833976601, -9961638803674197865477353330/258536940627643651)

Commentary

E has torsion Z/5Z and rank exactly 6. It is the Z/5 Tate normal form y² + (1−t)xy − ty = x³ − tx² at t = −89/175163. It was found in a sweep of the Kubert Z/5 family over t = a/b with max(|a|,|b|) ≤ 600000 (4.4×10¹¹ curves): a Mestre–Nagao screen in C, followed by a 2-Selmer rank computation and a 2-descent with point search in Magma. Rank 6 is certified by six independent points (Magma CertifyPoints) and 2-Selmer rank 6. It was confirmed independently in PARI/GP: ellrank [6,6], regulator 2280197.331, torsion Z/5. Search performed by Claude code Opus 5.5.

last edited by Lewis Combes at 2026-10-30 20:59:55 UTC · history