curve #4965

y² + xy = x³ + x² − 7406113067287861086412332x + 17174147099464253053852169594515299360

a-invariants: [1, 1, 0, -7406113067287861086412332, 17174147099464253053852169594515299360]
c-invariants: [355493427229817332147791961, -14838463093939780839232498159652327086685]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 100584989890605528503345116207040193702
discriminant (Δ): -101420312542914153169999262352206139313659870158390604964054447885193788783248
Faltings height: 13.4646
naive height: 184.9961
primes of bad reduction: 2, 3, 7, 11, 17, 19, 29, 31, 47, 61, 89, 179, 181, 397, 503, 601, 743, 787, 887
regulator: 75665324902784.82935730734509503648005401260351197888715623133095
submitted by: The Wren
submitted at: 2026-10-21 00:00:08 UTC
last updated: 2026-10-21 00:00:08 UTC

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

(-3169657488919, 2967205708271852970)
(-1257434182381, 4949613384940794544)
(710510444261, 3502956954468451710)
(10420571053340459/2809, 950394860711859351788313/148877)
(10397001896201, 32619992214022301194)
(-1902633633409, 4937379315458258500)
(7058524654492649/4096, 809211751309411613429461/262144)
(15823708986289/9, 83603095928414002030/27)
(25108204871059, 125139843067908369469)
(186471271241299/36, 2344217420276511373679/216)
(-3818749635810941/1089, 5651499035861997735040/35937)
(11891398194769759/900, 1273848857865808307237713/27000)
(642175411238671/729, 66263833021719644324756/19683)

Commentary

Obtained by a rational 2-isogeny from curve #4611, submitted by Andrii Prokhorenkov: https://elliptic-rank.icarm.cloud/curve/4611. The source commentary identifies X660 at T = 135/64. The source model is y^2 + x*y = x^3 + x^2 - 9682089825240363517751292*x + 11586062042591897085329683155674638800. Applied a rational degree-2 isogeny and converted the target to its minimal model y^2 + x*y = x^3 + x^2 - 7406113067287861086412332*x + 17174147099464253053852169594515299360. Thirteen 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 100584989890605528503345116207040193702. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 00:00:08 UTC · history