diff --git a/src/sage/schemes/elliptic_curves/ell_finite_field.py b/src/sage/schemes/elliptic_curves/ell_finite_field.py
index 50b0fb4727c..87fa0ec67b8 100755
--- a/src/sage/schemes/elliptic_curves/ell_finite_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_finite_field.py
@@ -1399,6 +1399,7 @@ def has_order(self, value, num_checks=8):
             True
             sage: E.has_order(N^2)
             True
+            sage: del E._order
             sage: E.has_order(N^2 + N)
             True
 
@@ -1419,7 +1420,7 @@ def has_order(self, value, num_checks=8):
         # orders So we go with computing directly instead.
         # In #38341, the bound has been increased to a large value (2^64), but
         # it should be decreased (to ~100) after bug #38617 is fixed.
-        if q <= 2**64:
+        if q <= 2**64 or hasattr(self, "_order"):
             return self.order() == value
 
         # This might be slow
@@ -3008,24 +3009,24 @@ def EllipticCurve_with_prime_order(N):
         sage: set_random_seed(1337)
         sage: for _, E in zip(range(3), EllipticCurve_with_prime_order(10^9 + 7)):
         ....:     print(E)
-        verbose 2 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163
+        verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163
         Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437
-        verbose 2 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667
+        verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667
         Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307
         Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307
         sage: set_verbose(4)
         sage: set_random_seed(1337)
         sage: for _, E in zip(range(3), EllipticCurve_with_prime_order(10^9 + 7)):
         ....:     print(E)
-        verbose 4 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-19
+        verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-19
         ...
-        verbose 4 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-163
-        verbose 2 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163
+        verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-163
+        verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-163
         Elliptic Curve defined by y^2 = x^3 + 265977778*x + 120868502 over Finite Field of size 1000041437
-        verbose 4 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-179
+        verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-179
         ...
-        verbose 4 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-667
-        verbose 2 (2866: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667
+        verbose 4 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Testing D=-667
+        verbose 2 (...: ell_finite_field.py, EllipticCurve_with_prime_order) Computing the Hilbert class polynomial H_-667
         Elliptic Curve defined by y^2 = x^3 + 689795416*x + 188156157 over Finite Field of size 999969307
         Elliptic Curve defined by y^2 = x^3 + 999178436*x + 900579394 over Finite Field of size 999969307