Examples of nontrivial Manin constants

**Earlier this was a section; I made it into a subsection,
since it is short. -AmodWe present two sets of examples in which the Manin constant
is not
.
**

**I rewrote this subsection, and kept a copy of William's
original version after it. Feel free to pick the one you like.
-AmodUsing results of [Kil02], Adam Joyce [Joy05] proves
that there is a new optimal quotient of
with Manin
constant
.
Joyce's methods also produce examples with Manin
constant
at levels
and
.
For the convenience of the reader, we breifly discuss his
example at level
.
There are exactly two
elliptic curves
and
of prime conductor
, and
as subvarieties of
, so
is an
optimal quotient of
attached to a saturated ideal
. If
is the newform corresponding to
, then one finds
that
, and so
. However
is not
in the image of
. Thus the Manin constant
of
is divisible by
.
**

**As another class of examples, one
finds by computation for each prime
that
does not leave
stable.
Theorem 3.5 (with
)
then implies that the Manin constant of
is divisible by
for these values of
.
**