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See Construct elliptic curves as Jacobians in the manual.

sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain()
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)

See Construct elliptic curves as Jacobians in the manual.

sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain()
phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)

See Construct elliptic curves as Jacobians in the manual.

sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)

Edit: in this way you certainly get a map, but not necessarily an isomorphism; see the other answer.