curve #4966

y² + xy + y = x³ − x² − 32994734426395246870623932x − 60932386453876175388400003571476827265

a-invariants: [1, -1, 1, -32994734426395246870623932, -60932386453876175388400003571476827265]
c-invariants: [1583747252466971849789948721, 52645581896156142398213704459080033525975]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 27184304298374298547706309213830981063446
discriminant (Δ): 694956736817299408067429326383068460972919112829012431387530210104430057408512
Faltings height: 13.6563
naive height: 187.8888
primes of bad reduction: 2, 3, 7, 17, 41, 43, 67, 103, 167, 173, 271, 379, 563, 1009, 1663, 2441, 2731, 55819
regulator: 14649220022338242.4198744795690636309866475645852846320972219875320
submitted by: The Wren
submitted at: 2026-10-21 00:02:20 UTC
last updated: 2026-10-21 00:02:20 UTC

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

(-2317778439227, 1758058966472598229)
(-2273680372487, 1527425288935926729)
(-2241920235435, -1330721469867587771)
(-2756557218435, -3012222052705715795)
(-3740230525065, -3186254293577319035)
(-4353384568875, -448576922213514011)
(9364545695793, -21244017753742387871)
(1564125061193395/169, -45371036011019714851757/2197)
(8361099982173, -15738589199893910771)
(174222113044646487/24649, 29717233130802590001828151/3869893)
(-308870127626495483/142129, 38174588944302898500270957/53582633)
(174796118487685/9, -2197519322114675843017/27)
(-87906325660431/25, -425924722738029283063/125)
(-15049904391565151/4225, -927248851929208965005487/274625)

Commentary

Obtained by a rational 2-isogeny from curve #4612, submitted by Andrii Prokhorenkov: https://elliptic-rank.icarm.cloud/curve/4612. The source commentary identifies X660 at T = 75/41. The source model is y^2 + x*y + y = x^3 - x^2 - 9587363242690973783330492*x + 10536060714562308983254191000903086975. Applied a rational degree-2 isogeny and converted the target to its minimal model y^2 + x*y + y = x^3 - x^2 - 32994734426395246870623932*x - 60932386453876175388400003571476827265. 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 27184304298374298547706309213830981063446. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 00:02:20 UTC · history