increase minimum degree for Îlu formulas to 9

This is mainly done to exclude curves over GF(3) with 7 rational points,
which cause trouble for Îlu as they don't have any points defined only
in a quadratic extension of the base field. Hence, the resulting isogeny
won't be defined over GF(3).

Since √élu is still much slower than Vélu for degrees this small, noone
will miss the functionality thus excluded.

Author: yyyyx4

src/sage/schemes/elliptic_curves/hom_velusqrt.py

@@ -756,7 +756,7 @@ class EllipticCurveHom_velusqrt(EllipticCurveHom):
     INPUT:
 
     - ``E`` -- an elliptic curve over a finite field
-    - ``P`` -- a point on `E` of odd order `\geq 5`
+    - ``P`` -- a point on `E` of odd order `\geq 9`
     - ``codomain`` -- codomain elliptic curve (optional)
     - ``model`` -- string (optional); input to
       :meth:`~sage.schemes.elliptic_curves.ell_field.compute_model`
@@ -884,8 +884,8 @@ def __init__(self, E, P, *, codomain=None, model=None, Q=None):
         self._P = self._pre_iso(P)
 
         self._degree = self._P.order()
-        if self._degree % 2 != 1 or self._degree < 5:
-            raise NotImplementedError('only implemented for odd degrees >= 5')
+        if self._degree % 2 != 1 or self._degree < 9:
+            raise NotImplementedError('only implemented for odd degrees >= 9')
 
         if Q is not None:
             self._Q = E(Q)
@@ -1208,11 +1208,11 @@ def _random_example_for_testing():
             E.short_weierstrass_model()
         except ValueError:
             continue
-        if E.cardinality() < 5:
+        if E.cardinality() < 9:
             continue
         A = E.abelian_group()
         ds = max(A.invariants()).prime_to_m_part(2).divisors()
-        ds = [d for d in ds if 5 <= d < 1000]
+        ds = [d for d in ds if 9 <= d < 1000]
         if ds:
             deg = choice(ds)
             break