EllipticCurveHom_fractional fails when the characteristic divides the denominator

[harrshita123]

Steps To Reproduce

Consider the fractional representation [p]/p of the identity morphism in characteristic p:

p = 5
E = EllipticCurve(GF(p), [1, 1])
phi = E.scalar_multiplication(p) / p

phi.degree()
phi == E.scalar_multiplication(1)
phi.scaling_factor()
phi.to_isogeny_chain()

The first two expressions confirm that phi is a degree-one morphism equal to the identity:

1
True

Expected Behavior

Since phi = [p]/p represents the identity morphism, its scaling factor should be 1, and to_isogeny_chain() should return a degree-one morphism representing the identity.

Methods implemented through to_isogeny_chain(), including rational_maps(), x_rational_map(), kernel_polynomial(), and formal(), should also work.

Actual Behavior

phi.scaling_factor() raises a ZeroDivisionError. Consequently, phi.to_isogeny_chain() and the methods depending on it also fail.

Additional Information

EllipticCurveHom_fractional.scaling_factor() currently computes

return self._phi.scaling_factor() / self._d

When the base field has characteristic p and p divides the denominator d, both the numerator scaling factor and d can become zero in the base field. For [p]/p, this attempts to evaluate 0/0, although the resulting morphism is the identity and has scaling factor 1.

This also affects to_isogeny_chain(), which calls self.scaling_factor() while selecting the final isomorphism.

The source currently contains a related FIXME noting that this can crash when p divides d.

Environment

• OS: macOS 26.5.2
• Sage Version: 10.10.beta5 / current develop

Checklist

• I have searched the existing open and closed issues for a matching bug report without success.
• I have read the documentation and troubleshooting guide.