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Consider the following snippet:

R.<x, y, z> = QQ[]; C = Conic(27628*(x^2 - y^2) - 81746*x*y  - z^2); print([C.has_rational_point(algorithm = "local", obstruction = True), C.has_rational_point(algorithm = "default", obstruction = True)]);

This returns [(True, None), (False, 17)].

Consider the following snippet:There seems to be an inconsistency.

Define a polynomial ring in three variables:

sage: R.<x, y, z> = QQ[]; QQ[]

Define a conic:

sage: C = Conic(27628*(x^2 - y^2) - 81746*x*y  - z^2); print([C.has_rational_point(algorithm = "local", obstruction = True), C.has_rational_point(algorithm = "default", obstruction = True)]);
z^2)

This returns Check whether the conic has rational points, using [(True, None), algorithm="local":

sage: C.has_rational_point(algorithm="local", obstruction=True)
(True, None)

Check whether the conic has rational points, using algorithm="default":

sage: C.has_rational_point(algorithm="default", obstruction=True)
(False, 17)]17)
.