curve #5089

y² = x³ + 172976235117310548x

a-invariants: [0, 0, 0, 172976235117310548, 0]
c-invariants: [-8302859285630906304, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 135695137938136502727507141369888
discriminant (Δ): -331237344997119260553454130311763417072176291282341888
Faltings height: 8.9590
naive height: 130.6894
primes of bad reduction: 2, 3, 7, 11, 17, 19, 23, 31, 43, 47, 107, 179
regulator: 128250274.35784558829848034060348460968737260603487785102
submitted by: The Wren
submitted at: 2026-10-21 02:28:31 UTC
last updated: 2026-10-21 02:28:31 UTC

Witness: 9 independent points log in to add more points →

(242422992, 7495355278152)
(2106388944, 98540034383736)
(2796217732, 149488596170452)
(6955902052, 581172572346652)
(103213058256, 33159310832434248)
(2854122601/16, 386827798481333/64)
(2956312737/256, 5791295442960783/4096)
(3833313561/4, 258721832172573/8)
(3833313561/16, 475472714165757/64)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (9,27,29,41), reduced by rational fourth-power scaling. Reference: "High rank elliptic curves with torsion group Z/(2Z)", Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Selected in a bounded tuple search using a small-prime sieve and height screening, following the search approach described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Nine rational points checked exactly; PARI/GP returned rank bounds [9,9]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:28:31 UTC · history