Shimura (Crelle 221, 1966) considers the elliptic curve $E:y^2+y=x^3-x^2$ (although he doesn't use this equation) of conductor $11$ whose associated modular form is
$$
q\prod\_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2=\sum\_{n=1}^{+\infty}c\_nq^n
$$
where $q=e^{2i\pi\tau}$ and $\tau$ is in the upper half of $\bf C$. For a prime $l$, he denotes by $K\_l$ the extension of $\bf Q$ obtained by adjoining the $l$-torsion points of $E$ and shows that if $l\in[7,97]$, then ${\rm Gal}(K\_l|{\bf Q})$ is isomorphic to ${\rm GL}\_2({\bf F}\_l)$. 


**Question.** Is ${\rm Gal}(K\_l|{\bf Q})$ now known to be isomorphic to ${\rm GL}\_2({\bf F}\_l)$ even for $l>97$ ?


Even if the faithful representation ${\rm Gal}(K\_l|{\bf Q})\rightarrow{\rm GL}\_2({\bf F}\_l)$ fails to be surjective for a few $l>97$, does the recent proof of Serre's modularity conjecture not imply the


**Statement**. For every prime $l>5$ and every prime $p\neq11,l$, the characteristic polynomial of ${\rm Frob}\_p$ (thought of as an element of ${\rm GL}\_2({\bf F}\_l)$) is
$\equiv X^2-c\_pX+p \pmod l$ ?


Shimura shows this only for $l\in[7,97]$. 


**Addendum.** (2010/07/24) Looking at Shimura's paper beyond the first page shows that he actually proves (Section 3) that the characteristic polynomial for the action of ${\rm Frob}\_p$ on the $l$-adic Tate module $T\_l(E)$ is 


$X^2-a\_pX+p\in{\bf Z}\_l[X]$ 


for all primes $l$ and $p\neq11, l$ and (Section 6) that $a\_p=c\_p$ for all $p\neq11$. And yes, he does use the Eichler-Shimura relation. 



Reciprocity law for number fields defined by torsion points of modular elliptic curves
The first question is answered in Serre's 
[[Propriétés galoisiennes des points d'ordre fini des courbes elliptiques,
*Invent. Math.* **15**:4 (1972) 259--331]](http://dx.doi.org10.1007/BF01405086), on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the mod-$\ell$ representation is not surjective. *Sage* can do that efficiently for a given curve.


For every prime $p$ different from $\ell$ and $11$, the characteristic polynomial of $\rm{Frob}\_p$ is indeed $T^2 - c\_p T +p$ in $ \mathbb F\_{\ell}[T]$. The isomorphism
$\rm{Gal}(K \_{\ell}/\mathbb Q) \to \rm{Aut}(E[\ell])=\rm{GL} \_2(\mathbb F \_{\ell})$ sends $\rm{Frob}\_p$ to the Frobenius endomorphism $\phi:E[\ell] \to E[\ell]$ on $E/\mathbb{F}\_p$.
Your $c\_p$ is the trace of $ \phi$ and $p$ is the determinant of it since the Eichler--Shimura relation shows that $c\_p$ is the Fourier coefficent of the associated modular form. See [this answer](https://mathoverflow.net/questions/19390/intuition-behind-the-eichler-shimura-relation/19399#19399) for why it is so.


