curve #4920

y² = x³ + 4837001583721743854150043401475x

a-invariants: [0, 0, 0, 4837001583721743854150043401475, 0]
c-invariants: [-232176076018643704999202083270800, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 29947627930786122521100129658764695285207276277281520681184800
discriminant (Δ): -7242836186496100273720517166994397525633436690183738858399022413901040873369236321253379000000
Faltings height: 16.6995
naive height: 223.5751
primes of bad reduction: 2, 3, 5, 7, 11, 23, 41, 61, 89, 101, 643, 1361, 3491, 6091, 87049
regulator: 7717123732098616060.793635884807856566151225963094948360605584594
submitted by: The Wren
submitted at: 2026-10-20 22:53:37 UTC
last updated: 2026-10-20 22:53:37 UTC

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

(723532334796347521, 615444540179803540505007106)
(30161693346058725025/9216, -199575122653204358488857069425/884736)
(47003939816091544225/1521, 323071340376439125134183927350/59319)
(1743863331494750193025/133956, -73854942262788893384042033343175/49027896)
(637104342661937846809/10816, -16092308821735297693709471163523/1124864)
(64484906527276225/81, 48110423901732271379731850/729)
(12501517469734225, -1419262576161730387327850)
(1945047604405633225/131044, 401957617431982694530200333475/47437928)
(49263183621520968583969/150822961, 74431065631631097247011992992176022/1852256784041)
(4299239441320355379961/714025, -300111625378230279725991037174834/603351125)
(4075898141679535155625/23454649, -3303544868133753276014850674719250/113590865107)
(207182880369703992601/21904, 3061712974109979622067116295549/3241792)
(211362237274132266522216025/229567473424, 7957511687308608836144479454192568520925/109993122676587968)

Commentary

Obtained by a rational 2-isogeny from curve #4776, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4776. The source commentary attributes the original curve y^2 = x^3 - 19348006334886975416600173605900*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), followed by the rational change of coordinates dividing x by 4 and y by 8. With A = -19348006334886975416600173605900, the combined map is X = (x^2+A)/(4*x), Y = y*(x^2-A)/(8*x^2), giving y^2 = x^3 + 4837001583721743854150043401475*x. 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 29947627930786122521100129658764695285207276277281520681184800. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:53:37 UTC · history