The following worked for me:

```
sage: E = EllipticCurve(GF(43), [0, 6])
....: [P.xy() for P in E if P.order() == 13]
....:
[(13, 15),
(13, 28),
(26, 9),
(26, 34),
(27, 9),
(27, 34),
(33, 9),
(33, 34),
(35, 15),
(35, 28),
(38, 15),
(38, 28)]
```

Also a slightly bigger case:

```
sage: E = EllipticCurve(GF(2027), [0, 1])
....: [P.xy() for P in E if P.order() == 13]
[(155, 48),
(155, 1979),
(579, 647),
(579, 1380),
(1304, 910),
(1304, 1117),
(1584, 379),
(1584, 1648),
(1732, 836),
(1732, 1191),
(1819, 992),
(1819, 1035)]
```

For a "much bigger curve" (i.e. with more rational points) some improvements are necessary.
For instance:

```
p = (3*10^7).next_prime()
E = EllipticCurve(GF(p), [0, 4])
ord = E.order()
print(f'E is the following curve:\n{E}')
print(f'E has order {ord} = {ord.factor()}')
print(f'E has the generator(s): {E.gens()}')
```

We get the following information on `E`

:

```
E is the following curve:
Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 30000001
E has order 30010071 = 3 * 7 * 13 * 37 * 2971
E has the generator(s): ((14044277 : 14356696 : 1),)
```

Now

```
[P.xy() for P in E if P.order() == 13]
```

takes a looong time. But we can immediately get the elements of order $13$ as follows:

```
G = E.gens()[0] # we already know there is one generator of order <ord>
n = ZZ(ord/13)
Q = n*G
[(k*Q).xy() for k in [1..12]]
```

This gives:

```
[(28289013, 19261067),
(11842435, 11155846),
(26389003, 19261067),
(5321986, 10738934),
(15676831, 11155846),
(2480735, 11155846),
(2480735, 18844155),
(15676831, 18844155),
(5321986, 19261067),
(26389003, 10738934),
(11842435, 18844155),
(28289013, 10738934)]
```

Check:

```
sage: E( (2480735, 18844155) ).order()
13
```

Welcome to Ask Sage! Thank you for your question!

Ideally, provide an example of defining all the pieces to get to the actual question.

Ok, then take say $E$ to be defined over $F_{43}$ by $y^2=x^3+6$. This is a very simple BLS6 curve (hence has embedding degree 6). It has a prime oder subgroup of order 13 and I want to list the entire 13-torsion.