I sometimes read $\int\_{E(\mathbf{R})} \frac{dx}{2y + a\_1x + a\_3}$ and sometimes $\int\_{E(\mathbf{R})} |\frac{dx}{2y + a\_1x + a\_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.


So what's the correct definition for the constant appearing in the BSD conjecture?



How is the period of an elliptic curve defined exactly?
The comments above give already the answer, but for the sake of completeness let us be a bit more precise.


Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element in the period lattice $\Lambda$. Then the conjecture of Birch and Swinnerton-Dyer predicts that


$$
\frac{L^{\*}(E,1)}{[E(\mathbb{R}):E(\mathbb{R})^{o}] \cdot \Omega^{+}} = \frac{\prod\_{p} c\_p \cdot | Sha| \cdot Reg}{| E(\mathbb{Q})\_{tors}|^2}
$$


The denominator on the left, where the index is the number of connected components of $E(\mathbb{R})$, can also be written as the *absolute value* of $\int\_{E(\mathbb{R})}\omega\_E$ where $\omega\_E$ is a invariant differential of a global minimal Weierstrass model.


A better way of formulating the conjecture especially if $E$ is no longer defined over $\mathbb{Q}$ but over an arbitrary global field was given Tate. (See for instance conjecture 2.1 in  [Dokchitser's paper](http://arxiv.org/abs/math/0610290) for a formulation). Since there are no global minimal models anymore one has to make either a conjecture that is invariant of the choice of a model or work with the Néron model.


