Notation:

• $E$ is a non-CM Elliptic curve over $\mathbb{Q}$.
• $p$ is an ordinary prime.
• $f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$.
• $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$.
• $G_S:= \mathrm{Gal}(\mathbb{Q}_S/{\mathbb{Q}})$, where $\mathbb{Q}_S$ - maximal unramified extension outside the set $S=\{\text{ bad primes of } E \} \cup\{ p, \infty \}$.

Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p:=\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \\ 0 & d \end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega \lambda_p^{-1}(\overline{a}_p) & 0 \\ 0 & \lambda_p(\overline{a}_p) \end{pmatrix}$, where $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}_p$, $\overline{a}_p \in \mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$, and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f: G_p \rightarrow \mathrm{GL}_2(\mathbb{Z}/p^n \mathbb{Z})$, where $\mathbb{Z}/p^n\mathbb{Z} (\mathbb{Z}/p^n\mathbb{Z})^2 \simeq E[p^n]$, (E[p^n])$, the $p^n$-torsion points of $E$, for some fixed $n \geq 2$? Is it possible to compute it (or atleast it's order) by using MAGMA/SAGE/PARI?

Notation:

• $E$ is a non-CM Elliptic curve over $\mathbb{Q}$.
• $p$ is an ordinary prime.
• $f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$.
• $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$.
• $G_S:= \mathrm{Gal}(\mathbb{Q}_S/{\mathbb{Q}})$, where $\mathbb{Q}_S$ - maximal unramified extension outside the set $S=\{\text{ bad primes of } E \} \cup\{ p, \infty \}$.

Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p:=\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \\ 0 & d \end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega \lambda_p^{-1}(\overline{a}_p) & 0 \\ 0 & \lambda_p(\overline{a}_p) \end{pmatrix}$, where $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}_p$, $\overline{a}_p \in \mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$, and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f: G_p \rightarrow \mathrm{GL}_2(\mathbb{Z}/p^n \mathbb{Z})$, where $\mathbb{Z}/p^n\mathbb{Z} \simeq E[p^n]$, the $p^n$-torsion points of $E$, for some fixed $n \geq 2$? Is it possible to compute it (or atleast it's order) by using MAGMA/SAGE/PARI?

Notation:

• $E$ is a non-CM Elliptic curve over $\mathbb{Q}$.
• $p$ is an ordinary prime.
• $f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$.
• $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$.
• $G_S:= \mathrm{Gal}(\mathbb{Q}_S/{\mathbb{Q}})$, where $\mathbb{Q}_S$ - maximal unramified extension outside the set $S=\{\text{ bad primes of } E \} \cup\{ p, \infty \}$.

Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p:=\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \\ 0 & d \end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega \lambda_p^{-1}(\overline{a}_p) & 0 \\ 0 & \lambda_p(\overline{a}_p) \end{pmatrix}$, where $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}_p$, $\overline{a}_p \in \mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$, and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f: G_p \rightarrow \mathrm{GL}_2(\mathbb{Z}/p^n \mathbb{Z})$, where $\mathbb{Z}/p^n\mathbb{Z} \simeq E[p^n]$, the $p^n$-torsion points of $E$, for some fixed $n \geq 2$? Is it possible to compute it by using MAGMA/SAGE/PARI?

Is it possible to compute the following using MAGMA/SAGE/PARI :

Notation:

• $E$ is a non-CM Elliptic curve over $\mathbb{Q}$.
• $p$ is an ordinary prime, .
• $f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$, E$.
• $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$.
• $G_S:= \mathrm{Gal}({\mathbb {Q}}_S/{\mathbb{Q}})$, mathrm{Gal}(\mathbb{Q}_S/{\mathbb{Q}})$, where ${\mathbb {Q}}_S$ \mathbb{Q}_S$ - maximal unramified extension outside the set $S={p$, $\mathrm{ bad}$ $\mathrm{primes}$ $\mathrm{ of}$ $ E$, $\infty$ $}$. S=\{\text{ bad primes of } E \} \cup\{ \infty \}$.

Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p:=\mathrm{Gal}(\overline{\mathbb{Q}}_p/{\mathbb{Q}}_p)$ G_p:=\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \\ 0 & d \end{pmatrix}$. end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega {{\lambda}_p}^{-1}(\overline{a}_p) \lambda_p^{-1}(\overline{a}_p) & 0 \\ 0 & {\lambda}_p(\overline{a}_p) \end{pmatrix}$. Wherelambda_p(\overline{a}_p) \end{pmatrix}$, where $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}_p$, $\overline{a}_p$ $\in$ $\mathbb{F}_p$ \overline{a}_p \in \mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$ f$, and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f$: $G_p$ $\rightarrow$ $\mathrm{GL}_2(\mathbb{Z}/p^n \rho_f: G_p \rightarrow \mathrm{GL}_2(\mathbb{Z}/p^n \mathbb{Z})$, where $\mathbb{Z}/p^n\mathbb{Z}$ $\simeq$ $E[p^n]$, \mathbb{Z}/p^n\mathbb{Z} \simeq E[p^n]$, the $p^n$-torsion points of $E$, for some fixed $n$ $\geq$ 2n \geq 2$? Is it possible to compute it by using MAGMA/SAGE/PARI?

Is it possible to compute the following using MAGMA/SAGE/PARI :

Notation:

$E$ is a non-CM Elliptic curve over $p$ is an ordinary prime, $f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$, $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$. $G_S:= $\mathrm{Gal}({\mathbb \mathrm{Gal}({\mathbb {Q}}_S/{\mathbb{Q}})$, where ${\mathbb {Q}}_S$ - maximal unramified extension outside the set \ $S={p,$ S={p$, $\mathrm{ bad}$ $\mathrm{primes}$ $\mathrm{ of}$ $ E$, $\infty}$.\Assume \infty$ $}$. Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p$:=$\mathrm{Gal}(\overline{\mathbb{Q}}p/{\mathbb{Q}}{p})$ G_p:=\mathrm{Gal}(\overline{\mathbb{Q}}_p/{\mathbb{Q}}_p)$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \ 0 & d \end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega {{\lambda}_p}^{-1}(\overline{a}_p) & 0 \ 0 & {\lambda}_p(\overline{a}_p) \end{pmatrix}$. Where, $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}p$, \overline{a}_p$, $\overline{a}{p}$ \overline{a}_p$ $\in$ $\mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$ and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f$: $G_p$ $\rightarrow$ $\mathrm{GL_2(\mathbb{Z}/p^n\mathbb{\Z}}),\ \mathrm{GL}_2(\mathbb{Z}/p^n \mathbb{Z})$, where $\mathbb{Z}/p^n\mathbb{\Z}$ \mathbb{Z}/p^n\mathbb{Z}$ $\simeq$ $E[p^n]$, the $p^n$-torsion points of $E$, for some fixed $n$ $\geq$ 2? Is it possible to compute it by using MAGMA/SAGE/PARI?

Is it possible to compute the following using MAGMA/SAGE/PARI :

Notation:

$E$ is a non-CM Elliptic curve over $p$ is an ordinary prime, \\$f$ $f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$,\\ E$, $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$.\\ f$. $\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$. $G_S:= $\mathrm{Gal}({\mathbb {Q}}_S/{\mathbb{Q}})$, where ${\mathbb {Q}}_S$ - maximal unramified extension outside the set\ $S={p,$ $\mathrm{ bad}$ $\mathrm{primes}$ $\mathrm{ of}$ $ E$, $\infty}$.\Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p$:=$\mathrm{Gal}(\overline{\mathbb{Q}}p/{\mathbb{Q}}{p})$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \ 0 & d \end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega {{\lambda}_p}^{-1}(\overline{a}_p) & 0 \ 0 & {\lambda}_p(\overline{a}_p) \end{pmatrix}$. Where, $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}p$, $\overline{a}{p}$ $\in$ $\mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$ and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f$: $G_p$ $\rightarrow$ $\mathrm{GL_2(\mathbb{Z}/p^n\mathbb{\Z}}),\ where $\mathbb{Z}/p^n\mathbb{\Z}$ $\simeq$ $E[p^n]$, the $p^n$-torsion points of $E$, for some fixed $n$ $\geq$ 2? Is it possible to compute it by using MAGMA/SAGE/PARI?

Is it possible to compute the following using MAGMA/SAGE/PARI :

Notation:

$E$ is a non-CM Elliptic curve over  $p$ is an ordinary prime, \\$f$ - cuspidal eigenform of weight $k$ = 2  attached with $E$,\\ $\rho_f$ - the global 2-dimensional $p$-adic Galois representation attached with $f$.\\ 

$\rho_f$ : $G_S$ $\rightarrow$ $\mathrm{GL}_2({\mathbb{Z}}_p)$. $G_S:= $\mathrm{Gal}({\mathbb {Q}}_S/{\mathbb{Q}})$, where ${\mathbb {Q}}_S$ - maximal unramified extension outside the set\ $S={p,$ $\mathrm{ bad}$ $\mathrm{primes}$ $\mathrm{ of}$ $ E$, $\infty}$.\Assume that the residual representation $\overline{\rho}_f$ is $p$-split.

The prime $p$ is an ordinary prime. So the the image of $\rho_f$ restricted to the decomposition group $G_p$:=$\mathrm{Gal}(\overline{\mathbb{Q}}p/{\mathbb{Q}}{p})$ will be of the form $\rho_f$ $\mid$ $G_p$ $\sim$ $\begin{pmatrix} a & * \ 0 & d \end{pmatrix}$. The residual representation $\overline{\rho}_f$ is $p$-split. So, $\overline{\rho}_f$ $\sim$ $\begin{pmatrix} \omega {{\lambda}_p}^{-1}(\overline{a}_p) & 0 \ 0 & {\lambda}_p(\overline{a}_p) \end{pmatrix}$. Where, $\lambda_p$ is an unramified character which sends $\mathrm{Frob}_p$ to $\overline{a}p$, $\overline{a}{p}$ $\in$ $\mathbb{F}_p$ is the mod $p$ reduction of the $p$-th coefficent $a_p$ of $f$ and $\omega$ is the $p$-adic cyclotomic character.

Question: What is the image of the representation $\rho_f$: $G_p$ $\rightarrow$ $\mathrm{GL_2(\mathbb{Z}/p^n\mathbb{\Z}}),\ where $\mathbb{Z}/p^n\mathbb{\Z}$ $\simeq$ $E[p^n]$, the $p^n$-torsion points of $E$, for some fixed $n$ $\geq$ 2? Is it possible to compute it by using MAGMA/SAGE/PARI?