sagemathgh-37967: Modify tests in ell_point.py to be more generic f…

…or Conway database update

    
As part of the Conway polynomials database update, two tests in
`schemes.elliptic_curves.ell_point` caused issues due to their
dependency on the precise implementation of the finite field
`GF(65537^2)`:
- The first test (of `.weil_pairing`) checks whether the Sage
implementation still works for an example of fixed `7282`-torsion points
- The second test (of `.tate_pairing`) checks whether the new
implementation, which uses PARI up to reduction of the pairing,
coincides with the old Sage implementation (still used e.g. in
SageMathCell, but not available in the current code anymore)

To help with the database update, we adapt these tests to not be
dependent on the exact minimal polynomial used for finite field
generation:
- We turn both tests into examples (with fixed moduli) for their
respective methods
- We modify the test of `.weil_pairing` to check for two randomly chosen
`7282`-torsion points whether `algorithm="sage"` and `algorithm="pari"`
yield the same result

Please let me know if you think other polynomial-independent tests are
better suited as replacements.

See sagemath/conway-polynomials#5 (especially [comment
10](https://github.com/sagemath/conway-
polynomials/pull/5#issuecomment-2101462886) and Update 2 in [comment
11](https://github.com/sagemath/conway-
polynomials/pull/5#issuecomment-2101498241)) for more details.

CC: @yyyyx4 @GiacomoPope @JohnCremona
    
URL: sagemath#37967
Reported by: Sebastian A. Spindler
Reviewer(s):

src/sage/schemes/elliptic_curves/ell_point.py

@@ -1818,6 +1818,17 @@ def weil_pairing(self, Q, n, algorithm=None):
             sage: z.multiplicative_order()
             360
 
+        Another larger example::
+
+            sage: F = GF(65537^2, modulus=[3,-1,1], name='a')
+            sage: F.inject_variables()
+            Defining a
+            sage: E = EllipticCurve(F, [0,1])
+            sage: P = E(22, 28891)
+            sage: Q = E(-93, 2728*a + 64173)
+            sage: P.weil_pairing(Q, 7282, algorithm='sage')
+            53278*a + 36700
+
         An example over a number field::
 
             sage: # needs sage.rings.number_field
@@ -1833,16 +1844,20 @@ def weil_pairing(self, Q, n, algorithm=None):
 
         TESTS:
 
-        Check that the original Sage implementation still works::
+        Check that the original Sage implementation still works and
+        that the result coincides with the PARI implementation::
 
             sage: # needs sage.rings.finite_rings
             sage: GF(65537^2).inject_variables()
             Defining z2
             sage: E = EllipticCurve(GF(65537^2), [0,1])
-            sage: P = E(22, 28891)
-            sage: Q = E(-93, 40438*z2 + 31573)
-            sage: P.weil_pairing(Q, 7282, algorithm='sage')
-            19937*z2 + 65384
+            sage: R, S = E.torsion_basis(7282)
+            sage: a, b = ZZ.random_element(), ZZ.random_element()
+            sage: P = a*R + b*S
+            sage: c, d = ZZ.random_element(), ZZ.random_element()
+            sage: Q = c*R + d*S
+            sage: P.weil_pairing(Q, 7282, algorithm='sage') == P.weil_pairing(Q, 7282, algorithm='pari')
+            True
 
         Passing an unknown ``algorithm=`` argument should fail::
 
@@ -2047,19 +2062,19 @@ def tate_pairing(self, Q, n, k, q=None):
             sage: Px.weil_pairing(Qx, 41)^e == num/den
             True
 
-        TESTS:
-
-        Check that the PARI output matches the original Sage implementation::
+        An example over a large base field::
 
-            sage: # needs sage.rings.finite_rings
-            sage: GF(65537^2).inject_variables()
+            sage: F = GF(65537^2, modulus=[3,46810,1], name='z2')
+            sage: F.inject_variables()
             Defining z2
-            sage: E = EllipticCurve(GF(65537^2), [0,1])
+            sage: E = EllipticCurve(F, [0,1])
             sage: P = E(22, 28891)
             sage: Q = E(-93, 40438*z2 + 31573)
             sage: P.tate_pairing(Q, 7282, 2)
             34585*z2 + 4063
 
+        TESTS:
+
         The point ``P (self)`` must have ``n`` torsion::
 
             sage: P.tate_pairing(Q, 163, 2)