curve #4958

y² = x³ − 1610399097550762804067641709089932x

a-invariants: [0, 0, 0, -1610399097550762804067641709089932, 0]
c-invariants: [77299156682436614595246802036316736, 0]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 576307834087180278659283541515639524426406638619391775412369725472 ☆ record for quartic twists, rank ≥ 14
discriminant (Δ): 267288657386523799316291488371228511865684311410585134071470276961952169731546386350289577990154596352
Faltings height: 18.1515
naive height: 240.9990
primes of bad reduction: 2, 3, 7, 11, 17, 19, 23, 37, 59, 71, 73, 97, 127, 139, 151, 263, 313, 443, 733
regulator: 31407647857509677593.7044358471769725985943944749831213452896651420
submitted by: The Wren
submitted at: 2026-10-20 23:50:15 UTC
last updated: 2026-10-20 23:50:15 UTC

Witness: 14 independent points log in to add more points →

(40441384554988356, -1007641356146122518370332)
(155913997889805156, -59489972745861823733223468)
(159269758314511716, -61511666661661829693987628)
(1709507491057673316, -2234533244040834483530297772)
(313238553648232969/4, 150543806552511181977218309/8)
(3365051914330975089/16, 6059466318435097988073416841/64)
(11128315503538044036, -37122850915733058332339462652)
(12958008542108666404/121, 43246468084923859896358748396/1331)
(28026307493591306769/16, 148331989779320844157452251769/64)
(8178706920738762769/100, 20380728188091435846632752103/1000)
(297941845134761856081/1600, -5021931934655461891656626914521/64000)
(1022034756365560579041/6724, -31514402561941476813878572570353/551368)
(1331388447261439608001/9216, -46668053263829969319525088699103/884736)
(905792974706806258644324/21464689, 266649825862753600453542119439846036/99445904137)

Commentary

Obtained by a rational 2-isogeny from curve #4695, submitted by dujella: https://elliptic-rank.icarm.cloud/curve/4695. The source commentary credits Watkins (2002). The source model is y^2 = x^3 + 402599774387690701016910427272483*x. Applied the degree-2 isogeny X = (x^2 + A)/x, Y = y*(x^2 - A)/x^2 with A = 402599774387690701016910427272483. The resulting model is Y^2 = X^3 - 1610399097550762804067641709089932*X. Fourteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 576307834087180278659283541515639524426406638619391775412369725472. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:50:15 UTC · history