I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of

D. Jeon, C. H. Kim, Y. Lee,

Families of elliptic curves over cubic number fields with prescribed torsion subgroups, Mathematics of Computation, V. 80, 273, January 2011, p. 579-591, JSTOR: 41104715.

My code snippet with the saved output for Magma Calculator online is available for download from MEGA.

I observe that the following triples of rational $t$-values produce curves with similar characteristics:

$$t_1=t$$ $$t_2=1-\frac{1}{t_1}=1-\frac{1}{t}$$ $$t_3=1-\frac{1}{t_2}=\frac{1}{1-t}$$ For a triple $t_1,t_2,t_3$, the three elliptic curves are different over three different cubic fields, with different discriminants and conductors.

But the ranks, $j$-invariants, and heights of all generators are the same (e.g., for rank $2$ there will be height $h_1$ for generators $g_{11}, g_{12}, g_{13}$ and height $h_2$ for generators $g_{21}, g_{22}, g_{23}$, where $g_{ik}$ is the $i$-th generator for the curve created using $t_k$).

It was also pretty straightforward to derive the formula for the $j$-invariant and see that it is always rational: $$j=\frac{(t^3-3t^2+1)^3(t^9-9t^8+27t^7-48t^6+54t^5-45t^4+27t^3-9t^2+1)^3}{(t^3-6t^2+3t+1)(t^2-t+1)^3(-1+t)^9t^9}$$ Magma is unable to check whether the curves are isomorphic, as they are defined over different cubic fields:

```
>> IsIsomorphic(E1, E2);
^
Runtime error in 'IsIsomorphic': Curves must be defined over the same base ring
```

Isogeneity check is unavailable over number fields (only over rationals or finite fields):

```
>> IsIsogenous(E1, E2);
^
Runtime error in 'IsIsogenous': Bad argument types
Argument types given: CrvEll[FldNum[FldRat]], CrvEll[FldNum[FldRat]]
```

The curves seem to be **essentially the same (not distinct in any real sense)** to me, even though they might not be considered isomorphic and/or isogenous.

Question 1:Does there exist a proper mathematical name for "essentially the same" used above, or the name for the observed connection between the curves?

Question 2:Is it possible to map a generator discovered on one of the curves to the other two curves? The explicit expression for the map is not a priority yet.

Question 3:If the height of the generator (but not the generator itself) is considered to be known, is it possible to speed up a search process for it? If so, how?

**Rationale for Questions 2 and 3:** It is very easy to determine both generators (with heights $1.798$ and $11.652$, default `Effort := 1`

in $45$ seconds) for $t_1=\frac{1}{5}$, harder to do so for $t_2=-4$ (`Effort := 1000`

helps, takes much longer), and very hard to recover the second generator for $t_3=\frac{5}{4}$ (`Effort := 1600`

fails).

z+t)/((z^2-t^2+t)*(z^2+tz-t^2+t)^2); c := -t*(z-t)*(z^2-tz+t)/((z^2-t^2+t)*(z^2+tz-t^2+t)); E1prime:=EllipticCurve([isom(1-c),isom(-b),isom(-b),0,0]); IsIsomorphic(E1prime,E2); $\endgroup$