calculating torsion points of elliptic curves through lattice

Question

Given an Elliptic Curve $E: y^2=x^3+ax+b$ in Weierstrass-form, I can transform it to $E': y^2=4x^3-g_2x-g_3$ and then I have an isomorphism: $C/L \simeq E'$ with $ z \mapsto (\wp(z),\wp'(z))$.
I can explicitly calculate the half-periods $\omega_1,\omega_2$. Now if I want to calculate a 2-Torsion point, the isomorhpism tells me that I can just calculate e.g. $a=\omega_1/2$ and then $P=(\wp(a),\wp'(a))$ should be a 2-Torsion point. I have tried this explicitly, but I didn't get proper results, i.e.the point $P$ doesn't have order 2. Did I make any mistake here or did I miscalculate at some point?

Answer 1

My answer is ... don't use Wolfram Alpha. It seems it's not computing things correctly.

Let $E$ be $y^2=x^3+x$. Then, Magma tells me that the periods for this curve are

$$[\omega_1,\omega_2]=[ 3.70814935460274383686770069439\cdots , 1.85407467730137191843385034719\cdots + (1.85407467730137191843385034719\cdots)\cdot i ]$$

Now we can use the map $\mathbb{C}\to E$ given by $[\wp,\wp']$ at the point $\omega_2/2$ and obtain the point:

$[ -1.00000000000000000000000000000\cdot i, 5.69951259140326437423671127619E-31 + 5.69951259140326437423671127619E-31\cdot i ]$

which is really close to $(-i,0)$. I used the following Magma code:

E:=EllipticCurve([0,0,0,1,0]);

Periods(E);

EllipticCurveFromPeriods(Periods(E));

EllipticExponential(E,Periods(E)[2]/2);

You can find the Magma documentation on this here.