bound for the rank of an elliptic curve

Theorem.

Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve given by the equation:

$$E:y^2=x(x-t)(x-s),\mathit{\text{ with }}t,s\in \mathbb{Z}$$

and suppose that $E$ has $s\mathrm{=}m\mathrm{+}a$ primes of bad reduction, with $m$ and $a$ being the number of primes with multiplicative and additive reduction respectively. Then the rank of $E$, denoted by $R_E$, satisfies:

$$R_E\le m+2a-1$$

Example.

As an application of the theorem above, we can prove that $E_1:y^2=x^3-x$ has only finitely many rational solutions. Indeed, the discriminant of $E_1$, $\mathrm{\Delta }=64$, is only divisible by $p=2$, which is a prime of (bad) multiplicative reduction. Therefore $R_{E_1}=0$. Moreover, the Nagell-Lutz theorem implies that the only torsion points on $E_1$ are those of order $2$. Hence, the only rational points on $E_1$ are:

$$\{𝒪,(0,0),(1,0),(-1,0)\}.$$