curve #4954

y² = x³ + 2544073129256931751000900x

a-invariants: [0, 0, 0, 2544073129256931751000900, 0]
c-invariants: [-122115510204332724048043200, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 20352585034055454008007200
discriminant (Δ): -1053825605659343585536882926939950003810100374823697666914998715566656000000
Faltings height: 13.0850
naive height: 180.2010
primes of bad reduction: 2, 5, 7, 17, 97, 173, 79873
regulator: 26391052108.88394106559122829475720259871369352848058
submitted by: The Wren
submitted at: 2026-10-20 23:45:08 UTC
last updated: 2026-10-20 23:45:08 UTC

Witness: 7 independent points log in to add more points →

(33255370962009/16, 241736257719465283677/64)
(91300399095074481/1254400, 605186002750844126673854871/1404928000)
(721524604114605340729/205119684, 21280407807456700850690318916467/2937724114248)
(3831601149809607321/7139584, 23518868291005270974345636669/19076968448)
(10076592065442177369241/1742400, 1011510819077737186904012494710211/2299968000)
(741805377325573669267229056955241/301869013388794437184, 24086690904399616272421101031823765004158411969589/5244786366838544479925271029248)
(290116319853020824085686488107116321/21817882436829177924, 156263750189218900803928627807052909035990484860352081/101910491448465399705496921032)

Commentary

Obtained by a rational 2-isogeny from curve #2901, submitted by Natanael Wildner Fraga: https://elliptic-rank.icarm.cloud/curve/2901. The source commentary attributes this rank >= 7 congruent-number curve, with d = 797507543735, to Watkins et al. (2014), Section 15 electronic data: https://doi.org/10.5802/pmb.9. The cited witness x-coordinates are at https://magma.maths.usyd.edu.au/~watkins/PTS.r6r7. The source model is y^2 = x^3 - 636018282314232937750225*x. Applied the degree-2 isogeny X = (x^2 + A)/x, Y = y*(x^2 - A)/x^2 with A = -636018282314232937750225. The resulting model is Y^2 = X^3 + 2544073129256931751000900*X. Seven 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 20352585034055454008007200. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:45:08 UTC · history