# Galois representation attached to $3$-torsion points of an elliptic curve

Let

$E$ - Elliptic curve defined over ${\mathbb{Q}}$.

$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ of $\mathbb{Q}$.

$E[3]$ - $3$-torsion points of $E$.

Suppose $\rho$ denotes the $G_{\mathbb{Q}}$-representation associated to $E[3]$. If $E$ has a $3$-torsion point over $\mathbb{Q}$, how to prove $$\rho \sim \left( {\begin{array}{cc} 1 & \eta \\ 0 & \chi \\ \end{array} } \right)$$ where $\chi$ is the mod 3 cyclotomic character $?$

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Assuming that $E$ has a rational $3$-torsion point, that point must be fixed, and hence your representation takes the form $$\rho \sim \left( \begin{matrix} 1 & \eta_{i} \\ 0 & \chi \end{matrix}\right)$$ as you say. It's a general fact that the determinant of the $G_{\mathbb{Q}}$ representation attached to an elliptic curve is the cyclotomic character. This follows because the Weil pairing is bilinear, alternating, and Galois-invariant. (See Proposition III.8.1 in Silverman's Arithmetic of Elliptic Curves.)
If $E$ has no rational $3$-torsion point, will $\rho$ still have the same representation $?$ – Robert May 22 '14 at 17:14