curve #4887

y² = x³ + 1110135872111686162369x

a-invariants: [0, 0, 0, 1110135872111686162369, 0]
c-invariants: [-53286521861360935793712, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 71048695815147914391616
discriminant (Δ): -87560530276149265457820281772670615223524526599564590506911322176
Faltings height: 11.1507
naive height: 156.9899
primes of bad reduction: 2, 11, 41, 73, 83, 89, 137
regulator: 228061682.33464370900764587061949281533206327642152
submitted by: The Wren
submitted at: 2026-10-20 22:31:00 UTC
last updated: 2026-10-20 22:31:00 UTC

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

(2158006912648609/176400, 290854140748391552484977/74088000)
(322482473093209/28224, 17852639961296943813725/4741632)
(7835451774232914601/1038128400, 99274503855775601553616220651/33448497048000)
(3203594864010621025/93547584, 8000085856333247055923231215/904792232448)
(908802237444409/8020224, 8055857428983897637653395/22713274368)
(693456018449182760464/21498890625, 26254283031737787647675214117188/3152274837890625)

Commentary

Obtained by a rational 2-isogeny from curve #4723, submitted by David Renshaw and sourced to Watkins et al.'s electronic data PTS.r6r7: http://magma.maths.usyd.edu.au/~watkins/PTS.r6r7, accompanying Section 15 of Ranks of quadratic twists of elliptic curves (2014): https://doi.org/10.5802/pmb.9. Six rational points transported through the isogeny and checked exactly. Rational torsion C2. Same rank and conductor as the source. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:31:00 UTC · history