How to Compute $E(\mathbb{Q})_{\tor}$

The following theorem yields an algorithm to compute $E(\mathbb{Q})_{\tor}$.

Theorem 3.1 (Nagell-Lutz)   Suppose that $y^2 = x^3 + ax + b$ (with $a,b\in\mathbb{Z}$) defines an elliptic curve $E$ over  $\mathbb{Q}$, let $\Delta = -16(4a^3 + 27b^2)$ be the discriminant, and suppose that $P=(x,y)\in E(\mathbb{Q})_{\tor}$. Then $x$ and $y$ are integers and either $y=0$, in which case $P$ has order $2$, or $y^2 \mid \Delta$.

Proof. [Non-proof] I will not prove this theorem. However, you can find a readable proof in Chapter II of Silverman and Tate's Rational Points on Elliptic Curves. $\qedsymbol$

Warning: Nagell-Lutz is NOT an if and only if statement. There are points of infinite order that satisfy the conclusion of Theorem 3.1. For example, the point $(1,3)$ on $y^2 = x^3 + 8$ has integer coordinates and $y^2 = 9 \mid \Delta = -16\cdot 27\cdot 3^2$. However,

$$(1,3) + (1,3) = \left( -\frac{7}{4}, -\frac{13}{8}\right).$$

Since the coordinates of $(1,3)+(1,3)$ are not integers, it follows from the contrapositive (not converse!) of Nagell-Lutz that $(1,3)$ must be a point of infinite order.

Example 3.2   The following is a list of elliptic curves with each possible torsion subgroup. Tom Womack (a graduate student in Nottingham, where Robin Hood lives) has a web page, http://www.tom.womack.net/maths/torsion.htm, which contains PARI code that lists infinitely many elliptic curve with each torsion subgroup.

Curve	$E(\mathbb{Q})_{\tor}$
$y^2 = x^3 - 2$	$\{0\}$
$y^2 = x^3 + 8$	$\mathbb{Z}/2\mathbb{Z}$
$y^2 = x^3 + 4$	$\mathbb{Z}/3\mathbb{Z}$
$y^2 = x^3 + 4x$	$\mathbb{Z}/4\mathbb{Z}$
$y^2 -y = x^3 - x^2$	$\mathbb{Z}/5\mathbb{Z}$
$y^2 = x^3 + 1$	$\mathbb{Z}/6\mathbb{Z}$
$y^2 = x^3 -43x + 166$	$\mathbb{Z}/7\mathbb{Z}$
$y^2 + 7xy = x^3 +16x$	$\mathbb{Z}/8\mathbb{Z}$
$y^2 + xy +y = x^3 - x^2 - 14x +29$	$\mathbb{Z}/9\mathbb{Z}$
$y^2 + xy = x^3 -45x + 81$	$\mathbb{Z}/10\mathbb{Z}$
$y^2 + 43xy - 210y = x^3 - 210x^2$	$\mathbb{Z}/12\mathbb{Z}$
$y^2 = x^3 - 4x$	$(\mathbb{Z}/2\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$
$y^2 = x^3 + 2x^2 - 3x$	$(\mathbb{Z}/4\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$
$y^2 + 5xy - 6y = x^3 - 3x^2$	$(\mathbb{Z}/6\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$
$y^2 + 17xy - 120y = x^3 -60x^2$	$(\mathbb{Z}/8\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$

The elltors function in PARI computes torsion subgroups:

? ?elltors
elltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, 
generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-Nagell.
? e=ellinit([17,-60,-120,0,0]);
? elltors(e)
%4 = [16, [8, 2], [[30, -90], [-40, 400]]]
? e.disc
%5 = 51438240000
? e.disc % 90^2        \\ verify Nagell-Lutz
%6 = 0
? e.disc % 400^2       \\ verify Nagell-Lutz
%7 = 0