The issue was observed when constructing an elliptic curve over an extension field, that is itself an extension of another finite field. The resulting elliptic curve is a general elliptic curve that has no .cardinality() method, while the expected type would be an elliptic curve over a finite field.
In the following examples, an elliptic curve defined by y^2 = x^3 + 1 is constructed over a finite field which is a 6-th degree extension of a 2-nd degree extension of F_11. More specifically, the field is (F_11[X]/(x^2+1))[Y]/(y^6 + (x+3)). The following code demonstrates the issue: F = GF(11) R.<x> = PolynomialRing(F,'x') F1.<x> = F.extension(x^2^1,'x') S.<y> = PolynomialRing(F1,'y') F2 = F1.extension(y^6 + (x+3),'y') E = EllipticCurve(F2,[0,1]) E.cardinality() The error message is: AttributeError: 'EllipticCurve_generic_with_category' object has no attribute 'cardinality' Another example resulting in a construction of a generic elliptic curve: F = GF(11) R.<x> = PolynomialRing(F,'x') F1.<x> = F.extension(x^2^1,'x') S.<y> = PolynomialRing(F1,'y') F2 = F1.extension(y^6 + (x+3),'y') E = EllipticCurve(F,[0,1]) E = E.base_extend(F1) E = E.base_extend(F2) E.cardinality() AttributeError: 'EllipticCurve_generic_with_category' object has no attribute 'cardinality' I copy John Cremona's analysis of what's possibly happening, following our discussion in PARI/GP user's mailing list, together with two examples provided by him. The first example results in a generic elliptic curve, while the second example constructs a proper elliptic curve over a finite field. The way you have constructed the field extensions has somehow bypassed Sage's normal way to construct field extensions, with the result that the final object E in your code has the wrong type (it should be EllipticCurve_finite_field or similar). There is no cardinality method for such a generic elliptic curve type. If it had the right type it would find the cardinality easily, by recognising that the j-invariant was in the prime field, doing a point count there (or in this case seeing that j=0 and using the appropriate formula) and then determining the carinality over the extension field using a standard formula. With apologies to pari people for more Sage code: sage: F = GF(11) sage: x = polygen(F) sage: F1.<a> = F.extension(x^2+1) sage: y = polygen(F1) sage: F2.<b> = F1.extension(y^6+a+3) sage: F, F1, F2 (Finite Field of size 11, Finite Field in a of size 11^2, Univariate Quotient Polynomial Ring in b over Finite Field in a of size 11^2 with modulus b^6 + a + 3) sage: E = EllipticCurve(F,[0,1]) sage: E1 = E.change_ring(F1) sage: E2 = E.change_ring(F2) sage: E.cardinality() 12 sage: E1.cardinality() 14 sage: E2.cardinality() (error message as in your post) If instead you construct F2 by giving just its degree over F1 and not a specific polynomial, all is well: sage: F2 = F1.extension(6) sage: F2 Finite Field in z12 of size 11^12 sage: E2 = E.change_ring(F2) sage: E2.cardinality() 3138424833600 but the polynomial is not the one you wanted: sage: F2.gen().minpoly() x^12 + x^8 + x^7 + 4*x^6 + 2*x^5 + 5*x^4 + 5*x^3 + 6*x^2 + 5*x + 2 -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/83aba583-60ad-4736-94dd-ff7fd63cccb3%40googlegroups.com.
