curve #1143

y² + xy = x³ − 17804316368161176920698170x − 28860668275885780962533093060587753500

a-invariants: [1, 0, 0, -17804316368161176920698170, -28860668275885780962533093060587753500]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 138952056675401970
discriminant (Δ): 1377479632666872463261645551040351718160097205332763153831191715029472774400
Faltings height: 13.3008
naive height: 186.0381
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 47, 59, 73, 347, 3467
regulator: 650507.429405225019947489897014689568671923173672
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:27 UTC
last updated: 2026-09-23 21:37:27 UTC

Witness: 5 independent points log in to add more points →

(-16604870452958076300/6754801, -3988277467568331091290844950/17555727799)
(-3955975540920654510/1615441, 477153039011132864705198790/2053225511)
(-2350103741370, 41575475331057360)
(-11660356860822915/4624, -10212576192621072614295/314432)
(-13952033926182072610974/5535508801, 21099728055008301932226264596682/411847390303201)

Commentary

Obtained by computing a rational isogeny from curve#1088 and mapping all 5 independent witness points to the global minimal model. Isogenies preserve the Mordell-Weil rank; the image points are independently certified by the site verifier. Torsion computed exactly with PARI/GP elltors. Conductor is unchanged by the isogeny.

last edited by David Renshaw at 2026-09-23 21:37:27 UTC · history