diff --git a/src/sage/schemes/elliptic_curves/ell_curve_isogeny.py b/src/sage/schemes/elliptic_curves/ell_curve_isogeny.py
index d03346deacc..1d8837356c1 100644
--- a/src/sage/schemes/elliptic_curves/ell_curve_isogeny.py
+++ b/src/sage/schemes/elliptic_curves/ell_curve_isogeny.py
@@ -1018,12 +1018,6 @@ def __init__(self, E, kernel, codomain=None, degree=None, model=None, check=True
         # Inheritance house keeping
         self.__perform_inheritance_housekeeping()
 
-        # over finite fields, isogenous curves have the same number
-        # of rational points, hence we copy over cached curve orders
-        if self.__base_field.is_finite():
-            self._codomain._fetch_cached_order(self._domain)
-            self._domain._fetch_cached_order(self._codomain)
-
     def _eval(self, P):
         r"""
         Less strict evaluation method for internal use.
@@ -1157,6 +1151,16 @@ def _call_(self, P):
             Traceback (most recent call last):
             ...
             TypeError: (20 : 90 : 1) fails to convert into the map's domain Elliptic Curve defined by y^2 = x^3 + 7*x over Number Field in th with defining polynomial x^2 + 3, but a `pushforward` method is not properly implemented
+
+        Check that copying the order over works::
+
+            sage: E = EllipticCurve(GF(431), [1,0])
+            sage: P, = E.gens()
+            sage: Q = 2^99*P; Q.order()
+            27
+            sage: phi = E.isogeny(3^99*P)
+            sage: phi(Q)._order
+            27
         """
         if P.is_zero():
             return self._codomain(0)
@@ -1187,7 +1191,16 @@ def _call_(self, P):
             yP = self.__posti_ratl_maps[1](xP, yP)
             xP = self.__posti_ratl_maps[0](xP)
 
-        return self._codomain(xP, yP)
+        Q = self._codomain(xP, yP)
+        if hasattr(P, '_order') and P._order.gcd(self._degree).is_one():
+            # TODO: For non-coprime degree, the order of the point
+            # gets reduced by a divisor of the degree when passing
+            # through the isogeny. We could run something along the
+            # lines of order_from_multiple() to determine the new
+            # order, but this probably shouldn't happen by default
+            # as it'll be detrimental to performance in some cases.
+            Q._order = P._order
+        return Q
 
     def __getitem__(self, i):
         r"""
diff --git a/src/sage/schemes/elliptic_curves/ell_field.py b/src/sage/schemes/elliptic_curves/ell_field.py
index 6b64df4d075..2b38144b046 100644
--- a/src/sage/schemes/elliptic_curves/ell_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_field.py
@@ -1024,53 +1024,6 @@ def division_field(self, l, names='t', map=False, **kwds):
         else:
             return L
 
-    def _fetch_cached_order(self, other):
-        r"""
-        This method copies the ``_order`` member from ``other``
-        to ``self`` if their base field is the same and finite.
-
-        This is used in :class:`EllipticCurveIsogeny` to keep track of
-        an already computed curve order: According to Tate's theorem
-        [Tate1966b]_, isogenous elliptic curves over a finite field
-        have the same number of rational points.
-
-        EXAMPLES::
-
-            sage: E1 = EllipticCurve(GF(2^127-1), [1,2,3,4,5])
-            sage: E1.set_order(170141183460469231746191640949390434666)
-            sage: E2 = EllipticCurve(GF(2^127-1), [115649500210559831225094148253060920818, 36348294106991415644658737184600079491])
-            sage: E2._fetch_cached_order(E1)
-            sage: E2._order
-            170141183460469231746191640949390434666
-
-        TESTS::
-
-            sage: E3 = EllipticCurve(GF(17), [1,2,3,4,5])
-            sage: hasattr(E3, '_order')
-            False
-            sage: E3._fetch_cached_order(E1)
-            Traceback (most recent call last):
-            ...
-            ValueError: curves have distinct base fields
-
-        ::
-
-            sage: E4 = EllipticCurve([1,2,3,4,5])
-            sage: E4._fetch_cached_order(E1.change_ring(QQ))
-            sage: hasattr(E4, '_order')
-            False
-        """
-        if hasattr(self, '_order') or not hasattr(other, '_order'):
-            return
-        F = self.base_field()
-        if F != other.base_field():
-            raise ValueError('curves have distinct base fields')
-        if not F.is_finite():
-            raise ValueError('base field must be finite')
-        n = getattr(other, '_order', None)
-        if n is not None:
-            self._order = n
-
     def isogeny(self, kernel, codomain=None, degree=None, model=None, check=True, algorithm=None):
         r"""
         Return an elliptic-curve isogeny from this elliptic curve.
diff --git a/src/sage/schemes/elliptic_curves/ell_finite_field.py b/src/sage/schemes/elliptic_curves/ell_finite_field.py
index 31e193a64b6..a8235585b17 100644
--- a/src/sage/schemes/elliptic_curves/ell_finite_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_finite_field.py
@@ -1289,6 +1289,46 @@ def set_order(self, value, num_checks=8):
 
         self._order = value
 
+    def _fetch_cached_order(self, other):
+        r"""
+        This method copies the ``_order`` member from ``other`` to
+        ``self``. Both curves must have the same finite base field.
+
+        This is used in
+        :class:`~sage.schemes.elliptic_curves.hom.EllipticCurveHom`
+        to keep track of an already computed curve order: According
+        to Tate's theorem [Tate1966b]_, isogenous elliptic curves
+        over a finite field have the same number of rational points.
+
+        EXAMPLES::
+
+            sage: E1 = EllipticCurve(GF(2^127-1), [1,2,3,4,5])
+            sage: E1.set_order(170141183460469231746191640949390434666)
+            sage: E2 = EllipticCurve(GF(2^127-1), [115649500210559831225094148253060920818, 36348294106991415644658737184600079491])
+            sage: E2._fetch_cached_order(E1)
+            sage: E2._order
+            170141183460469231746191640949390434666
+
+        TESTS::
+
+            sage: E3 = EllipticCurve(GF(17), [1,2,3,4,5])
+            sage: hasattr(E3, '_order')
+            False
+            sage: E3._fetch_cached_order(E1)
+            Traceback (most recent call last):
+            ...
+            ValueError: curves have distinct base fields
+        """
+        if hasattr(self, '_order') or not hasattr(other, '_order'):
+            return
+        F = self.base_field()
+        if F != other.base_field():
+            raise ValueError('curves have distinct base fields')
+        n = getattr(other, '_order', None)
+        if n is not None:
+            self._order = n
+
+
 # dict to hold precomputed coefficient vectors of supersingular j values (excluding 0, 1728):
 
 supersingular_j_polynomials = {}
diff --git a/src/sage/schemes/elliptic_curves/ell_generic.py b/src/sage/schemes/elliptic_curves/ell_generic.py
index 154a581bc2a..4cd98d31e00 100644
--- a/src/sage/schemes/elliptic_curves/ell_generic.py
+++ b/src/sage/schemes/elliptic_curves/ell_generic.py
@@ -559,6 +559,8 @@ def __call__(self, *args, **kwds):
                       (ell_point.EllipticCurvePoint_field,
                        ell_point.EllipticCurvePoint_number_field,
                        ell_point.EllipticCurvePoint)):
+            if P.curve() is self:
+                return P
             # check if denominator of the point contains a factor of the
             # characteristic of the base ring. if so, coerce the point to
             # infinity.
diff --git a/src/sage/schemes/elliptic_curves/hom.py b/src/sage/schemes/elliptic_curves/hom.py
index bdf2c969079..e6806b7868e 100644
--- a/src/sage/schemes/elliptic_curves/hom.py
+++ b/src/sage/schemes/elliptic_curves/hom.py
@@ -35,6 +35,39 @@ class EllipticCurveHom(Morphism):
     """
     Base class for elliptic-curve morphisms.
     """
+    def __init__(self, *args, **kwds):
+        r"""
+        Constructor for elliptic-curve morphisms.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve(GF(257^2), [5,5])
+            sage: P = E.lift_x(1)
+            sage: E.isogeny(P)                        # indirect doctest
+            Isogeny of degree 127 from Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field in z2 of size 257^2 to Elliptic Curve defined by y^2 = x^3 + 151*x + 22 over Finite Field in z2 of size 257^2
+            sage: E.isogeny(P, algorithm='factored')  # indirect doctest
+            Composite morphism of degree 127 = 127:
+              From: Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field in z2 of size 257^2
+              To:   Elliptic Curve defined by y^2 = x^3 + 151*x + 22 over Finite Field in z2 of size 257^2
+            sage: E.isogeny(P, algorithm='velusqrt')  # indirect doctest
+            Elliptic-curve isogeny (using Îlu) of degree 127:
+              From: Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field in z2 of size 257^2
+              To:   Elliptic Curve defined by y^2 = x^3 + 119*x + 231 over Finite Field in z2 of size 257^2
+            sage: E.montgomery_model(morphism=True)   # indirect doctest
+            (Elliptic Curve defined by y^2 = x^3 + (199*z2+73)*x^2 + x over Finite Field in z2 of size 257^2,
+             Elliptic-curve morphism:
+               From: Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field in z2 of size 257^2
+               To:   Elliptic Curve defined by y^2 = x^3 + (199*z2+73)*x^2 + x over Finite Field in z2 of size 257^2
+               Via:  (u,r,s,t) = (88*z2 + 253, 208*z2 + 90, 0, 0))
+        """
+        super().__init__(*args, **kwds)
+
+        # Over finite fields, isogenous curves have the same number of
+        # rational points, hence we copy over the cached curve orders.
+        if isinstance(self.base_ring(), finite_field_base.FiniteField) and self.degree():
+            self._codomain._fetch_cached_order(self._domain)
+            self._domain._fetch_cached_order(self._codomain)
+
     def _repr_type(self):
         """
         Return a textual representation of what kind of morphism
diff --git a/src/sage/schemes/elliptic_curves/hom_composite.py b/src/sage/schemes/elliptic_curves/hom_composite.py
index b2096eda5b8..7ca458dc43d 100644
--- a/src/sage/schemes/elliptic_curves/hom_composite.py
+++ b/src/sage/schemes/elliptic_curves/hom_composite.py
@@ -398,6 +398,16 @@ def _call_(self, P):
             sage: R = E.lift_x(15/4 * (a+3))
             sage: psi(R)    # indirect doctest
             (1033648757/303450 : 58397496786187/1083316500*a - 62088706165177/2166633000 : 1)
+
+        Check that copying the order over works::
+
+            sage: E = EllipticCurve(GF(431), [1,0])
+            sage: P, = E.gens()
+            sage: Q = 2^99*P; Q.order()
+            27
+            sage: phi = E.isogeny(3^99*P, algorithm='factored')
+            sage: phi(Q)._order
+            27
         """
         return _eval_factored_isogeny(self._phis, P)
 
diff --git a/src/sage/schemes/elliptic_curves/hom_velusqrt.py b/src/sage/schemes/elliptic_curves/hom_velusqrt.py
index 144b5400970..8f0befbc5ca 100644
--- a/src/sage/schemes/elliptic_curves/hom_velusqrt.py
+++ b/src/sage/schemes/elliptic_curves/hom_velusqrt.py
@@ -896,24 +896,23 @@ def __init__(self, E, P, *, codomain=None, model=None, Q=None):
         if codomain is not None and model is not None:
             raise ValueError('cannot specify a codomain curve and model name simultaneously')
 
-        try:
-            self._raw_domain = E.short_weierstrass_model()
-        except ValueError:
-            raise NotImplementedError('only implemented for curves having a short Weierstrass model')
-        self._pre_iso = E.isomorphism_to(self._raw_domain)
-
         try:
             P = E(P)
         except TypeError:
             raise ValueError('given kernel point P does not lie on E')
-        self._P = self._pre_iso(P)
-
-        self._degree = self._P.order()
+        self._degree = P.order()
         if self._degree % 2 != 1 or self._degree < 9:
             raise NotImplementedError('only implemented for odd degrees >= 9')
 
+        try:
+            self._raw_domain = E.short_weierstrass_model()
+        except ValueError:
+            raise NotImplementedError('only implemented for curves having a short Weierstrass model')
+        self._pre_iso = E.isomorphism_to(self._raw_domain)
+        self._P = self._pre_iso(P)
+
         if Q is not None:
-            self._Q = E(Q)
+            self._Q = self._pre_iso(E(Q))
             EE = E
         else:
             self._Q = _point_outside_subgroup(self._P)  # may extend base field
diff --git a/src/sage/schemes/elliptic_curves/weierstrass_morphism.py b/src/sage/schemes/elliptic_curves/weierstrass_morphism.py
index a9ad45f6fb2..85c673101d5 100644
--- a/src/sage/schemes/elliptic_curves/weierstrass_morphism.py
+++ b/src/sage/schemes/elliptic_curves/weierstrass_morphism.py
@@ -634,12 +634,28 @@ def __call__(self, P):
             (-3/4 : 3/4 : 1)
             sage: w(P).curve() == E.change_weierstrass_model((2,3,4,5))
             True
+
+        TESTS:
+
+        Check that copying the order over works::
+
+            sage: E = EllipticCurve(GF(431^2), [1,0])
+            sage: i = next(a for a in E.automorphisms() if a^2 == -a^24)
+            sage: P,_ = E.gens()
+            sage: P._order
+            432
+            sage: i(P)._order
+            432
+            sage: E(i(P))._order
+            432
         """
         if P[2] == 0:
             return self._codomain(0)
-        return self._codomain.point(baseWI.__call__(self,
-                                                          tuple(P._coords)),
-                                          check=False)
+        res = baseWI.__call__(self, tuple(P._coords))
+        Q = self._codomain.point(res, check=False)
+        if hasattr(P, '_order'):
+            Q._order = P._order
+        return Q
 
     def __invert__(self):
         r"""