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Alternate possibility : PD.get_point() does accept rational coordinates :

sage: p = PD.get_point(1/7+I/2); p
Point in PD 1/2*I + 1/7

Can you try to determine p's exact coordinates (possibly by other means) ? I mean coordinates in QQbar.

Using the current data, you can try to convert (a bit awkwardly) from the CC-derived representation :

sage: pprime=PD.get_point((lambda a,b:a+I*b)(*[f(p.coordinates()).exact_rational() for f in (real, imag)])) ; pprime
Point in PD 2196861306417441/4503599627370496*I + 2489029731445931/18014398509481984