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Real periods of elliptic curves over $\mathbb{Q}$ of rank $3$

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Numbers

$N$

$c_4$, $c_6$

$\omega_1$

INPUT{numbers.yaml} (not shown in preview)

Definition

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 $3$ 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..10000])
if E.rank() == 3]
```

Links

Data properties

Entries are of type: real number

Table is complete: no

Sources of data: [1] (list of curves with bounded conductor)