curve #4918

y² = x³ + 375691491394896586916214665028x

a-invariants: [0, 0, 0, 375691491394896586916214665028, 0]
c-invariants: [-18033191586955036171978303921344, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 31365354823671479158815749670876719302406866939637832497952
discriminant (Δ): -3393704716368552479990825843551330631783388798661900488380557118471138205010290482694524928
Faltings height: 16.0607
naive height: 215.9093
primes of bad reduction: 2, 3, 11, 13, 29, 31, 43, 47, 59, 79, 83, 173, 313, 373, 1487, 3457
regulator: 1256929426486255360.537298287201529229121430741895798343250649853
submitted by: The Wren
submitted at: 2026-10-20 22:52:31 UTC
last updated: 2026-10-20 22:52:31 UTC

Witness: 13 independent points log in to add more points →

(63016352864656, -4891313663433900760472)
(16405596103886224, -2102765885235988356634264)
(91094570804206637056/442225, -2729242154387650557652657767296/294079625)
(713400755650969/196, -3208826168567475874703011/2744)
(54766595564524321/196, 30898029785886176769990737/2744)
(8167519884446224/9, 890723253669188662757384/27)
(27637297950664249/16, -4875243635568196832873491/64)
(159943032271718656/729, 189803873200090995956836448/19683)
(2001968622036496/9, 262574895009664781267192/27)
(27188222681034686209/27556, 166894096665028594493841213601/4574296)
(69688329427062784/9, 18454254189831162575346304/27)
(196316738772296704/488601, -132693237385765051998757472576/341532099)
(13490311365409161031620496/31562720281, -86625888245138825327413929876094697048/5607401322402179)

Commentary

Obtained by a rational 2-isogeny from curve #4774, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4774. The source commentary attributes the original curve y^2 = x^3 - 93922872848724146729053666257*x to Aguirre, Castañeda and Peral, High rank elliptic curves with torsion group Z/(2Z), Mathematics of Computation 73 (2004), 323–331: https://doi.org/10.1090/S0025-5718-03-01547-3. Applied the standard degree-2 isogeny with kernel generated by (0,0), sending y^2 = x^3 + A*x to y^2 = x^3 - 4*A*x. Here A = -93922872848724146729053666257, giving y^2 = x^3 + 375691491394896586916214665028*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2. Thirteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion C2. Same rank and conductor as the source; conductor 31365354823671479158815749670876719302406866939637832497952. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:52:31 UTC · history