# Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?

Consider the ordinary elliptic curves $$E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1$$ over the field $$\mathbb{F}_2$$. They are quadratic twists to each other. I checked that the Kummer surface of $$E \!\times\! E^\prime$$, i.e., the quotient $$E \!\times\! E^\prime/[-1]$$ has the affine model $$K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2,$$ where $$y := x_1y_2 + y_1x_2$$.

Let $$t := x_1$$, $$x := x_2$$. It is almost obvious that the elliptic curve $$K_t/\mathbb{F}_2(t)$$ is reduced (over $$\mathbb{F}_2(t)$$) to the form $$\mathcal{E}\!:y^2 + txy = x^3 + t(t^2 + t + 1)x^2 + t^4x.$$

Is there a way to find any $$\mathbb{F}_2(t)$$-point on $$\mathcal{E}$$ outside $$\mathcal{E}[2] = \{(0:0:1), (0:1:0)\}?$$

• I did some computations with magma and it seems this curve has rank 0 (and torsion of order 2). So they are probably the only two points. – Xarles Aug 26 '19 at 17:27
• @Xarles, what are your computations? Magma does not work with elliptic K3 surfaces. – Dima Koshelev Aug 26 '19 at 19:03
• I used TwoIsogenySelmerGroups, which gives a bound for the rank of 0. See magma.maths.usyd.edu.au/magma/handbook/text/1472 – Xarles Aug 27 '19 at 9:58

Any $$\mathbb F_2(t)$$-point of $$K_t$$ would give, upon pullback to $$E \times E'$$, a $$\mathbb F_2(E)$$-point of $$E'$$. Because $$E$$ is ordinary, $$a_2(E)\neq 0$$, hence $$E$$ is not isogenous to its quadratic twist $$E'$$, so any such point arises from an $$\mathbb F_2$$-point of $$E'$$. Because $$E'$$ has two $$\mathbb F_2$$-points, these are the only ones.