Re: h_x of points on a rank-11 elliptic curve

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

Charles

I have E as [0, 0, 1, 0, 
44182596082121121317135170025680399046545625711306] not as given in your 
post

I had to create E from 5 points in P, successfully recovering the curve

I did reduce all 11 points to

[[-30156002278649820, 4093799681127459731025817], [11364087102067560, 
6756491872572362690626342], [22027709433525510, 
7407488162428235233522657], [50205120192991089/4, 
54352836820882753760852333/8], [-17812608047383695/4, 
53122823763086349528835211/8], [11724873521668020, 
6767159346634715672034457], [-138658831412368575/4, 
12719819443574268333325811/8], [994768217796990, 
6647073075327662243966017], [532896351059436225/16, 
576457310785324883248677823/64], [31523799036070860, 
8689612672637879411112217], [-35000505136795350, 1142689668110264775353617]]
? ellheights(E,%47)
 = [14.994742445428104708894826619224684865, 
15.324869012650739859192571311623802138, 
15.614650786757431382861235106402970368, 
15.803592607069303005618880725252017512, 
16.718982789104867986553669905935870803, 
17.464420459195601109014170439745218235, 
17.692215164960348268258086502880736253, 
17.894913288775905975805182678959806575, 
18.363955447008100324359296591108849148, 
18.513920310364999118122034009188675982, 
18.554764480419613116956240480557588323]

and the original heights of P are

[14.994742445428104708894826619224684865, 
15.324869012650739859192571311623802138, 
15.614650786757431382861235106402970368, 
15.803592607069303005618880725252017512, 
16.718982789104867986553669905935870803, 
16.974552891939263692172763088520846586, 
17.464420459195601109014170439745218235, 
17.692215164960348268258086502880736253, 
17.831522940445636165665877817614765512, 
17.894913288775905975805182678959806575, 
18.363955447008100324359296591108849148]

I also noticed that 76.61/4 = 19.1525 which is > 18.3639 for P and for 
my last point height 18.5547...

I am not sure either of what is going on.

Randall

On 5/2/24 21:04, Charles Greathouse wrote:
> I'm trying to work my way through the paper
> https://arxiv.org/abs/2403.17955
> and I'm at Proposition 2.1.
>
> I have initialized the elliptic curve as
> m0=13293998056584952174157235; E=ellinit([0,-432*m0]);
>
> I tried to use the rational points found in
> https://arxiv.org/abs/math/0403116
> where the curve is apparently defined as
> E=ellinit([0,1,0,0,44182596082121121317135170025680399046545625711306]);
> and its independent points as
> Pvec=[[-30156002278649820, 
> 4093799681127459731025817],[11364087102067560, 
> 6756491872572362690626342],[-20835788771691894, 
> 5927660006237675713476241],[1134264920569989390, 
> 1208031685828825118221478017],[8907565209691176834, 
> 26585114133655761890666064910],[111849199886121334, 
> 37992674604901443769570910],[11724873521668020, 
> 6767159346634715672034457],[-138658831412368575/4, 
> 12719819443574268333325811/8],[165971060901522240, 
> 67941788876402816577138982],[994768217796990, 
> 6647073075327662243966017],[532896351059436225/16, 
> 576457310785324883248677823/64]]
> but I can't replicate the result
> max{h_x(P_i) | 1 ≤ i ≤ 11} = 76.61
> and so must be doing something (several things?) wrong.