Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\omega_2$ be two periods obtained by integrating eta along a basis for the first singular homology of $E(\mathbb{C})$. Let $\Lambda$ be the lattice generated by $\omega_1$ and $\omega_2$. Is it necessarily the case that $g_2(\Lambda)$ and $g_3(\Lambda)$ lie in $K$, and does someone have a reference?
Edit Some background (hopefully I haven't messed up varying notations for theta functions)
If we scale $\Lambda$ by some $\lambda \in \mathbb{C}^*$ then $g_2(\lambda \Lambda) = \lambda^{-4} g_2(\Lambda)$ and $g_3( \lambda \Lambda) = \lambda^{-6} g_3(\Lambda)$.
Since $E/K$ has an equation of the form $y^2 = x^3 - Ax - B$ with $A,B \in K$, we may assume that it has an equation of the form $y^2 = 4x^3 - ax - b$ with $a,b \in K$.
On p485 of Whittaker and Watson it is asserted that if the Weierstrass equation is $y^2 = 4x^3 - a x - b = 4(x-e_1)(x-e_2)(x-e_3)$ (where $e_1, e_2, e_3$ are chosen so that $(e_1-e_2)/(e_1-e_3)$ is not a real number greater than $1$) then there there exist $\omega_1^\prime$ and $\omega_2^\prime$ for $E$ such that $(e_1 - e_2) = \omega_1^{\prime 2} \pi^2 \vartheta_4^4(\tau)$. Here $\tau = \omega_1^\prime/\omega_2^\prime$, and the $\omega_i^\prime$ are chosen so that $\tau$ has positive imaginary part, which places a restriction on the choice of basis for the singular homology in my question. (Of course, since they were writing in 1927, they provide no discussion of how the $\omega_i^\prime$ are defined relative to $K$.)
Moreover $x$ as in 3 as a function of $z \in \mathbb{C}$ is given by $$x(z) = \omega^{\prime 2}_1 \frac{\vartheta_2^2(\omega_1^\prime z ;\tau)}{\vartheta_1^2(\omega_1^\prime z ; \tau)} \vartheta_3^2(0 ;\tau) \vartheta_4^2(0;\tau) +e_1$$
So my question is really just: assuming a good choice of first homology basis are $\omega_1$ and $\omega_2$ as in the question the same as the $\omega_1^\prime$ and $\omega_2^\prime$ of Watson and Whittaker?
In "Sur l’équation fonctionnelle de la fonction thêta de Riemann" Moret-Bailly asserts that the section $$2 \pi \sqrt{-1} \vartheta_4(0;\tau)^8 (dz_1)^4$$ is a trivialising section of the line bundle $$\omega^4 \otimes TH^{-8}$$ on the moduli space of elliptic curves over $\mathbb{Z}[1/2]$. (Here $\omega$ is the Hodge bundle and $TH$ is the bundle of theta nulls).
Somehow 3 and 6 ought to be combined to show that the lattice of Whittaker and Watson is the same as the one generated by the algebraic periods in my question. I don't know if my question is really as difficult as all this, but I'm sure something like this must be in the literature already.
Further Edit In fact 3 and the answer to my question ought to follow from 6. The basic idea is that the value for the theta null given in 3 comes from the divisor of the theta function on the elliptic curve. Using the Koecher principle one ought to be able to show that Moret-Bailly's theorem implies that integrating the invariant differential gives the periods in 3, and this answers 5 in the affirmative.
