Are these two curves birationally equivalent? I'm trying to work out if the following two curves are birationally equivalent:
$$Y^2 = 2X^4 + 17X^2 + 12$$
$$2Y^2 = X^4 - 17$$
(I'm considering the above as the affine shorthands for the projective curves they represent)
I saw elsewhere online that I could use the following code in magma:
K<x,y>:=AffineSpace(Rationals(),2);
C1A:=Curve(K,2*x^4+17*x^2+12-y^2);
C2A:=Curve(K,x^4-17-2*y^2);
C1:=ProjectiveClosure(C1A);
C2:=ProjectiveClosure(C2A);
IsIsomorphic(C1,C2);

However this gives me a runtime error.
Can anyone see an error with my code/approach above? Alternatively, if there's another way I can check if two curves like the above are birationally equivalent either by hand or by using Magma / Sage or something similar that would be really helpful.

Smooth projective curves are birationally equivalent if and only if they are isomorphic (a rational map between smooth projective curves is automatically a morphism) - so we just want to find a smooth curve birationally equivalent to each $C_i$. The answer above is fine too, but I thought I'd continue your attempt at finding one with Magma.
I claim the curves are not birationally equivalent, so extending the field is fine (I will go to $L = \mathbb{Q}(\sqrt{-2}, \sqrt{31})$ ). I want to extend the base field simply to allow us to find some smooth points on $C_1$ and $C_2$ (PointSearch fails to find a smooth point up to height 10,000), at which point the intrinsic Magma function can form Elliptic Curves (it returns a smooth model $E_i$ over $L$, and a birational map from $C_i$ to $E_i$). Then we have
K<a> := QuadraticField(-2);
L<b> := PolynomialRing(K);
L<b> := ext<K | b^2 - 31>; 

A2<x,y> := AffineSpace(L, 2);

C1A := Curve(A2, 2*x^4 + 17*x^2 + 12 - y^2);
C2A := Curve(A2, x^4 - 17 - 2*y^2);

C1 := ProjectiveClosure(C1A);
C2 := ProjectiveClosure(C2A);

assert Genus(C1) eq 1;
assert Genus(C2) eq 1;

P1 := C1![1,b,1];
P2 := C2![1,2*a,1];

assert IsNonsingular(P1);
assert IsNonsingular(P2);

E1 := EllipticCurve(C1, P1);
E2 := EllipticCurve(C2, P2);

assert jInvariant(E1) ne jInvariant(E2);


which proves that $j(E_1) \neq j(E_2)$, so $E_1$ and $E_2$ are not isomorphic, and $C_1$ and $C_2$ are not birationally equivalent.
