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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?

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    It should work, but certainly it is not the best way of computing 2-torsion points, whose $x$-coordinates are simply the roots of $x^3+ax+b$... – 2017-01-31
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    I know that there are more simple/nice ways, yet I just wanted to see if it works the other way round. I guess I will need to revise what I have done again then, if the idea itself is not wrong – 2017-01-31
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    It should work. Can you add more detail about how you explicitly calculate the periods, and how you calculate the values of the Weierstrass functions? – 2017-01-31
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    Yes, sure: Let's explicitly look at $y'^2=x^3+x$, then I get with $y=y'/2$ as transformation: $y^2=4x^3+4x$. Now for this curve $g_2=-4$ and $g_3=0$.. Using WeierstrassHalfPeriods[{-4, 0}] in wolframalpha, I get $w_1=0.927037 - 0.927037 i$ and $\bar{w_1}$ as half periods. Now I calculate WeierstrassP[$w_1/2$, {-4, 0}]. I expect it to be either 0, i or-i, but it is neither. – 2017-01-31
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    Ah! Besides my answer below, it seems that we have missed the fact that the command you are using is "Weierstrass Half Period", so it seems to be giving you the half of a period! If you type WeierstrassP[0.927037−0.927037i, {-4, 0}] then you get i (or something very close to i). – 2017-02-03

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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.