bound for the rank of an elliptic curve
Theorem.
Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve^{} given by the equation:
$$E:{y}^{2}=x(xt)(xs),\mathit{\text{with}}t,s\in \mathbb{Z}$$ 
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:
$${R}_{E}\le m+2a1$$ 
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 NagellLutz theorem implies that the only torsion points on ${E}_{1}$ are those of order $2$. Hence, the only rational points on ${E}_{1}$ are:
$$\{\mathcal{O},(0,0),(1,0),(1,0)\}.$$ 
References

1
James Milne, Elliptic Curves, online course notes.
http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
Title  bound for the rank of an elliptic curve 

Canonical name  BoundForTheRankOfAnEllipticCurve 
Date of creation  20130322 14:24:25 
Last modified on  20130322 14:24:25 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  6 
Author  alozano (2414) 
Entry type  Theorem 
Classification  msc 14H52 
Related topic  ArithmeticOfEllipticCurves 