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2016-10-28 14:58:32 -0500 asked a question Elliptic curve defined with parameter

For example, I want to create and study the curve y^2 = x^3 + (u)x^2 + (16*u)x over finite fields. This curve parameterizes all elliptic curves with rational 2-torsion subgroups. I get errors when I try to use "u" when defining the curve. Is it possible to define a curve in this way and study it as a family of curves?

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2016-04-26 16:52:43 -0500 asked a question How to construct random divisor once the set of rational points over the Jacobian of a hyperelliptic curve is created?

I know how to deliberately create a divisor with knowledge of the mumford coordinates, but is there a way to generate a divisor randomly inside of X = J(FF) and extract its mumford coordinates as polynomials over FF?

Where divisors D = X([a,b]) are elements of X = J(FF), a and b are the mumford polynomials, and J is the Jacobian of a hyperelliptic curve.