Interesting results using a NF for isogenous curves to determine possible regulator heights

[American Citizen <website.reader3@gmail.com>]

Hello all:

I used a number field NF=nfinit(a^2 - 5879094002709584654188067784965) 
for an elliptic curve (1)

(1)   [1, 0, 0, -1473670506991667240419073314040, 
487857899055649012085793427889299493072289600]

which has 8 isogenous curves.

Looking at the torsion points for (1) I selected x = 990529152804881 
which gave the point p1 on the curve

(2)  p1 = [990529152804881, Mod(1/2*a - 990529152804881/2, a^2 - 
5879094002709584654188067784965)]

Using the command: vector(8,i,ellheight(ellinit(k[i],NF),p1)) where 
k[1..8] is the isogenous curve for (1)  I was able

to create a height matrix of point p1 on all 8 curves (assuming that I 
didn't make a mistake)

#  height

1  69.462661097568452675010358103720853484
2  68.178801124270948311447647167267142033
3  72.004467091893413430245291754771035103
4  69.211374519516486801909902481728475617
5  71.248351898944318245005414336630115533
6  70.848955458688340426113509363205938483
7  71.076540610040093288152888807161926145
8  73.995499888892081143756102359798632965

This roughly corresponds to the L-series computations of regulator 
height + SHA and proves to me that my idea is valid. Curve #2 has the 
lowest regulator height and is the one to use for point searching.

= Isogenous Curves: L-series-Reg  SHA
= 
[1,0,0,-1473670506991667240419073314040,487857899055649012085793427889299493072289600] 
586.0042
= 
[1,0,0,-1350882210032036138970816514040,604255810745174408864225385407951158557249600] 
293.0021
= 
[1,0,0,-21613375759952551922130559714040,38675151136497269167819387070355078625642209600] 
37504.2711
= 
[1,0,0,-76802094602026958682816514040,11216788650446842210221197792454992157249600] 
2344.0169
= 
[1,0,0,3910584245041201096206610543960,3229366054013312573785301403224849373420037200] 
18752.1356
= 
[1,0,0,-8822538010378633200216865972040,-9703116604031639452280171990115551185285698000] 
18752.1356
= 
[1,0,0,4438136821871996939736684043660,-36328127880290265739543247394077097481682050860] 
2400273.3547
= 
[1,0,0,-139665092896820718696935098515740,-635300473525359445488968869790600609112629919540] 
2400273.3547

It should be noted that 5 of these 8 curves probably have SHA > 1, 
especially #3, but height=72+ gives us a big clue.

Randall