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Note that the problem is not in defining the elliptic curve, but only when plotting it:

sage: E.change_ring(F)
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: E.plot()
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: E.change_ring(F).plot()
OverflowError: Python int too large to convert to C long

Note that the problem is not in defining the elliptic curve, but only when plotting it:

sage: E.change_ring(F)
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: E.plot()
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: E.change_ring(F).plot()
OverflowError: Python int too large to convert to C long

More precisely, the problem comes from the fact that the plot function iterates over the points of the curve:

sage: E = E.change_ring(F)
sage: E.points()
---------------------------------------------------------------------------
OverflowError                             Traceback (most recent call last)
<ipython-input-39-e168c98b87f8> in <module>()
----> 1 E.points()

/opt/sagemath/sage-source/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_finite_field.pyc in points(self)
    206         k = self.base_ring()
    207         if k.is_prime_field() and k.order()>50:
--> 208             v = self._points_via_group_structure()
    209         else:
    210             v =self._points_fast_sqrt()

/opt/sagemath/sage-source/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_finite_field.pyc in _points_via_group_structure(self)
    148         if ngens == 0:    # trivial group
    149             return H0
--> 150         for m in range(1,ni[0]):
    151             H0.append(H0[-1]+pts[0])
    152         if ngens == 1:    # cyclic group
OverflowError: Python int too large to convert to C long

Even if the loop was tuned so that it accepts such a large number (e.g. using xsrange instead of range), you will not get an error but, you will have to wait forever, there are just too many points:

sage: E.cardinality()
115792089237316195423570985008687907852837564279074904382605163141518161494337

So, you can think of making you own partial plot. Unfortunately, it seems currently not possible to look at the "first" points of the curve, since the iteration first computes all points:

sage: E.__iter__??
Source:   
    def __iter__(self):
        """
        Return an iterator through the points of this elliptic curve.

        EXAMPLES::

            sage: E = EllipticCurve(GF(11), [1,2])
            sage: for P in E:  print("{} {}".format(P, P.order()))
            (0 : 1 : 0) 1
            (1 : 2 : 1) 4
            (1 : 9 : 1) 4
            (2 : 1 : 1) 8
            ...
            (10 : 0 : 1) 2
        """
        for P in self.points():
            yield P

So, you can think of sampling some points using:

sage: E.random_element()
(5931825134738693147239085942431521905946451730071919123281591598082826125500 : 12327482641317859093710230640295762484447995871249429156192043444179262968436 : 1)