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Real periods of elliptic curves over $\mathbb{Q}$ of rank $2$
The real period $\omega_1$ of an elliptic curve over $\mathbb{Q}$ can be defined as $\Omega_E$ if $E(\mathbb{R})$ is connected and $\Omega_E/2$ if $E(\mathbb{R})$ has two components, where $\Omega_E = \int_{E(\mathbb{R})} |\omega|$, $E(\mathbb{R})$ is the set of real points of $E$, and $\omega$ is the invariant differential $dx/(2y+a_1 x+a_3)$ of a globally minimal model of $E$. This table lists $\omega_1$ for elliptic curves over $\mathbb{Q}$ of rank $2$ for the first few conductors.
Parameters
$N$
— conductor of $E$
$c_4$
— invariant $c_4$ of a minimal model of $E$
$c_6$
— invariant $c_6$ of a minimal model of $E$
Comments
(1)
Note that some authors refer to $\Omega$ as the real period.
(2)
The period lattice $\Lambda$ of $E$ is the lattice of periods of $\omega$. In terms of $\Lambda$, $\omega_1$ is the smallest positive real number in $\Lambda$.
(3)
The Birch and Swinnerton-Dyer conjecture [2] states that $\frac{L^{(r)}(E,1)}{r!} = \frac{|\text{Sha}(E)|\Omega_E R_E \prod_{p|N} c_p}{|E_{\text{Tor}}|^2}$.
(4)
The conductor $N$ can be deduced from $c_4$ and $c_6$. We display it for convenience.
(5)
A model of $E/\mathbb{Q}$ is given by $E: y^2 = x^3 - 27 c_4 x - 54 c_6$.
Programs
(P1)
Sage
numbers = [E.period_lattice().real_period()
for E in cremona_curves([1..1000])
if E.rank() == 2]
