Elliptic \(\operatorname{mod} \ell\) Galois representations which are not minimally elliptic

Abstract: In a recent preprint, F. Calegari has shown that for

e l l = 2, 3, 5

and 7 there exist 2-dimensional surjective representations

h o

of

with values in

F_{e} l l

coming from the

e l l

-torsion points of an elliptic curve defined over

Q

, but not minimally, i.e., so that any elliptic curve giving rise to

h o

has prime-to-

e l l

conductor greater than the (prime-to-

e l l

) conductor of

h o

. In this brief note, we will show that the same is true for any prime

e l l > 7

, concretely, we will show that for any such

e l l

the elliptic curve

E^ell: qquad Y^2 = X (X- 3^ell ) (X - 3^ell - 1)

is semistable, has bad reduction at 3, the associated

m o d e l l

Galois representation

h o

is surjective, unramified at 3, and there is no elliptic curve with good reduction at 3 whose associated

m o d e l l

representation is isomorphic to

h o

.

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