finding a point on the elliptic curve with parameter

edit

Try this:

sage: F = QQ['t'].fraction_field()
sage: t = F.gen()
sage: E = EllipticCurve(F, [t^4, t + t^3])

The elliptic curve functionality is restricted on general fields, for instance:

sage: R.<t> = QQ[]
sage: F = R.fraction_field()
sage: E = EllipticCurve(F, [1, t + t^3])
sage: E.point( (-t,0) )
(-t : 0 : 1)
sage: P = E.point( (-t,0) )
sage: P.is_finite_order()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-39-27a368cc0c53> in <module>()
----> 1 P.is_finite_order()
::::: many lines
NotImplementedError: Computation of order of a point not implemented over general fields.

I took above a simpler curve with an obvious torsion point.

Even the method is_finite_order failed to be executed.

Although in the above case we can at least see that $2P$ is $0$, if we ask for it:

sage: 2*P
(0 : 1 : 0)
sage: 2*P == E(0)
True

So if we get problems with the method is_finite_order, asking for the rank and/or for generators...

We can still use sage (and the own devices) to study the posted curve. After a quick pessimistic search of a polynomial $x=x(t)$ (divided by a power of $t$ maybe), such that $x^3 + t^4x + (t^3+t)$ should at least have an even order of $t$, i've got an optimistic point of view, that there should be no point on the curve $$ y^2 = x^3 + t^4x + (t^3+t)\ . $$ Why? Since if there would be one, say $(x,y) = (\ P_x(t)/Q_x(t)\ ,\ P_y(t)/Q_y(t)\ )$, then using specialization (not everywhere defined) morphisms $\mathbb{Q}(t)\to\mathbb{Q}$, $t\to t_0$ we would get some rational points on the elliptic curves defined over $\mathbb Q$, $$ y^2 = x^3 + t_0^4x + (t_0^3+t_0)\ . $$ Unless there is a division by zero in the one or the other denominator, when specializing $t$ to $t_0$. But even so, the denominators would have been already very complicated, since...

# for t0 in [ -29..29 ]:
for t0 in [ -25, -24, -20, -8, -3, -2, 2, 8, 12, 18, 22, 24, 29 ]:
    try:
        E = EllipticCurve( QQ, ( t0^4, t0^3 + t0 ) )
        ETP = E.torsion_points()
        if len( ETP ) > 1:
            print "... t0 = %3s | rank = ? | torsion points = %s" % ( t0, ETP )
            continue
        r = E.rank()
        if r > 0:
            print "... t0 = %3s | rank = %s | torsion points = %s" % ( t0, r, ETP )
            continue
        print "*** t0 = %3s | rank = %s | torsion points = %s" % ( t0, r, ETP )
    except Exception:
        pass

And we get:

*** t0 = -25 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 = -24 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 = -20 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  -8 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  -3 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  -2 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =   2 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =   8 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  12 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  18 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  22 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  24 | rank = 0 | torsion points = [(0 : 1 : 0)]
*** t0 =  29 | rank = 0 | torsion points = [(0 : 1 : 0)]

So an "unexpected" $\mathbb{Q}(t)$-rational point on $E$ is comming - in view of our first (naive) sieve - with denominators containing the factor $$\small Q=(t+25)(t+24)(t+20)(t+8)(t+3)(t+2)(t-2)(t-8)(t-12)(t-18)(t-22)(t-24)(t-29)\ .$$ (In fact $Q^2$ and $Q^3$ in $x$ and $y$.) This is not much, from an algorithmic point of view, but a good start.

For a family with one generating point $A(x(t),y(t))$, we can still use the specialization to "guess" $A$.