1 | initial version |

I am not sure about your question (do not hesitate to edit it to make it more precise), but you can define your elliptic cuve and points on a different field as follows:

```
sage: F = QQ[sqrt(33)]
sage: F
Number Field in sqrt33 with defining polynomial x^2 - 33
sage: F.inject_variables()
Defining sqrt33
sage: E = EllipticCurve(F,[0,0,0,-3267,45630])
sage: E
Elliptic Curve defined by y^2 = x^3 + (-3267)*x + 45630 over Number Field in sqrt33 with defining polynomial x^2 - 33
sage: P = [21+6*sqrt33, 54-42*sqrt33]
sage: P in E
True
sage: F = QQbar
sage: F
Algebraic Field
sage: F.inject_variables()
Defining I
sage: E = EllipticCurve(F,[0,0,0,-3267,45630])
sage: E
Elliptic Curve defined by y^2 = x^3 + (-3267)*x + 45630 over Algebraic Field
sage: P = [15+36*I,216-324*I]
sage: P in E
True
```

Note that by default, your elliptic curve is defined over the field of rationals:

```
sage: E = EllipticCurve([0,0,0,-3267,45630])
sage: E
Elliptic Curve defined by y^2 = x^3 - 3267*x + 45630 over Rational Field
```

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