question on mapping points from an elliptic curve back to a quartic

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

Suppose we consider a quartic with rational points

  Q(x,y) : -x^4 + 39/380*x^3 + 39/380*x + 1 = y^2

Some small sized rational points on the quartic have been found:

  qpts = [[0, 1], [0, -1], [20/19, 0], [-19/20, 0], [39/760, 
579121/577600], [39/760, -579121/577600], [364980/536731, 
271505562700/288080166361], [364980/536731, -271505562700/288080166361]]

An elliptic curve was derived from Q(x,y), the Weierstrass format is

  E = [0, 0, 0, 46908801/144400, 0]

Running the ellrank(E) command shows that this is a rank=3 curve and a 
Mordell-Weil basis was found

  p = [[6849/361, 15636267/137180], [68490000/17497489, 
50745080927115/1390647933253], [91918208204904964/97434336836799169, 
101298745897513561302124420799/5778586934871496820571475070]]

Using elliptic curve addition (and subtraction) on p, we create a pool 
of points on E <= specified height = 40. There are 34 points listed here.

  L = [[6849/361, -15636267/137180], [6849/361, 15636267/137180], 
[6849/400, -15636267/152000], [6849/400, 15636267/152000], 
[13689/577600, -1218027213/438976000], [13689/577600, 
1218027213/438976000], [2316484/169, -669881422241/417430], 
[2316484/169, 669881422241/417430], [8982009/2496400, 
-2612663608113/74941928000], [8982009/2496400, 
2612663608113/74941928000], [68490000/17497489, 
-50745080927115/1390647933253], [68490000/17497489, 
50745080927115/1390647933253], [182547121/2371600, 
-48129058505411/69393016000], [182547121/2371600, 
48129058505411/69393016000], [3614294161/40030929, 
-4431498926709179/5065513755660], [3614294161/40030929, 
4431498926709179/5065513755660], [119840302161/1444000000, 
-42453334703624409/54872000000000], [119840302161/1444000000, 
42453334703624409/54872000000000], [278122281129/65899510681, 
-12866633479133410977/338339949748176580], [278122281129/65899510681, 
12866633479133410977/338339949748176580], [9638599042404/135448289089, 
-5865110374832581310811/9471393633875468030], 
[9638599042404/135448289089, 
5865110374832581310811/9471393633875468030], 
[927685331970561/203214148302400, 
-115079925712041339103359/2896882712578168768000], 
[927685331970561/203214148302400, 
115079925712041339103359/2896882712578168768000], 
[8247603837763881/533864176461001, 
-23018645373929855138594961/246703985261676701330020], 
[8247603837763881/533864176461001, 
23018645373929855138594961/246703985261676701330020], 
[10128630871416609/481682221507600, 
-25508871036915992526709413/200860494103292894344000], 
[10128630871416609/481682221507600, 
25508871036915992526709413/200860494103292894344000], 
[13798538712239364/140920807194049, 
-313140159087921787500818271/317845658842779147095170], 
[13798538712239364/140920807194049, 
313140159087921787500818271/317845658842779147095170], 
[91918208204904964/97434336836799169, 
101298745897513561302124420799/5778586934871496820571475070], 
[965166608472041601/290919694852878400, 
-5237853648125655251289953601/156913274355282463802048000], 
[965166608472041601/290919694852878400, 
5237853648125655251289953601/156913274355282463802048000], 
[4570527917244381695586369/13272989264788276801600, 
-9784612672972695108233608237020790497/1529160653323709166957592608064000]]

We obtained an mapping for the quartic Q, i.e. [E,f,g] where g maps the 
points from L back to points on the quartic (quartic_to_ellmap() command)

Running the inverse map "g" on the 34 points in L results in the 
following 34 return values:

  [[]~, []~, []~, []~, [[0, 1], [39/760, -579121/577600]]~, [[0, -1], 
[39/760, 579121/577600]]~, []~, []~, []~, []~, []~, []~, []~, []~, []~, 
[]~, []~, []~, []~, []~, []~, []~, []~, []~, []~, []~, []~, []~, []~, 
[]~, []~, []~, []~, []~]

Question:

  Why are most of the points in the elliptic curve pool L unmappable 
back to Q(x,y)? This is surprising to me, as I believed that all the 
rational points on E were mappable back to Q(x,y)?

  I found by experience that I had to increase the height 10x in 
creating the pool to recover the qpts points listed above. (and yes, a 
pool of higher height points in L seems to have fully recovered qpts 
plus more points)

  I am a bit surprised to find that only 2 of the 34 points were 
mappable back to Q(x,y).