Appendix by J. Cremona: Verifying that

**Let
be a normalised rational newform for
. Let
be its period lattice; that is, the lattice of periods of
over
.
**

**We know that
is an elliptic curve
defined
over
and of conductor
. This is the optimal quotient of
associated to
. Our goal is two-fold: to identify
(by giving an explicit Weierstrass model for it with integer
coeffients); and to show that the associated Manin constant for
is
. In this section we will give an algorithm for this; our
algorithm applies equally to optimal quotients of
.
**

**As input to our algorithm, we have the following data:
**

- a -basis for , known to a specific precision;
- the type of the lattice (defined below); and
- a complete isogeny class of elliptic curves of conductor , given by minimal models, all with .

**So
is isomorphic over
to
for a unique
.
**

**The justification for this uses the full force of the modularity of
elliptic curves defined over
: we have computed a full set of
newforms
at level
, and the same number of isogeny classes of
elliptic curves, and the theory tells us that there is a bijection
between these sets. Checking the first few terms of the
-series
(i.e., comparing the Hecke eigenforms of the newforms with the traces
of Frobenius for the curves) allows us to pair up each isogeny class
with a newform.
**

**We will assume that one of the
, which we always label
, is
such that
and
(the period lattice of
)
are approximately equal. This is true in practice, because our method
of finding the curves in the isogeny class is to compute the
coefficients of a curve from numerical approximations to the
and
invariants of
; in all cases these are very close
to integers which are the invariants of the minimal model of an
elliptic curve of conductor
, which we call
. The other
curves in the isogeny class are then computed from
. For the
algorithm described here, however, it is irrelevant how the curves
were obtained, provided that
and
are
close (in a precise sense defined below).
**

**Normalisation of lattices: every lattice
in
which
defined over
has a unique
-basis
,
satisfying one of the following:
**

**Type 1:**and are real and positive; or**Type 2:**and are real and positive.

**For
we know the type from modular symbol calculations, and
we know
to a certain precision by numerical
integration; modular symbols provide us with cycles
such that the integral of
over
give
.
**

**For each curve
we compute (to a specific precision) a
-basis
for its period lattice
using the standard AGM method.
Here,
is the lattice of periods of the Néron
differential on
. The type of
is determined by the
sign of the discriminant of
: type
for negative discriminant,
and type
for positive discriminant.
**

**For our algorithm we will need to know that
and
are approximately equal. To be precise, we know that they
have the same type, and also we verify, for a specific postive
, that
**

and

**Pulling back the Néron differential on
to
gives
where
is the Manin constant
for
. Hence
**

- identify , to find which of the is (isomorphic to) the ``optimal'' curve ; and
- determine the value of .

**Our main result is that
and
, provided that the precision
bound
in (*) is sufficiently small (in most cases,
suffices). In order to state this precisely, we need
some further definitions.
**

**A result of Stevens says that in the isogeny class there is a curve,
say
, whose period lattice
is contained in
every
; this is the unique curve in the class with minimal
Faltings height. (It is conjectured that
is the
-optimal curve, but we do not need or use this fact. In
many cases, the
- and
-optimal curves are
the same, so we expect that
often; indeed, this holds for
the vast majority of cases.)
**

**For each
, we know therefore that
and also
. Let
be the
maximum of
and
.
**

**
**

Firstly,

Then

Hence

If , then , contradiction. Hence , so . Similarly, we have

and again we can conclude that , and hence .

Thus , from which the result follows.

**
**

**
**

**Finally, we give a slightly weaker result for
; in
this range we do not know
precisely, but only its
projection onto the real line. (The reason for this is that we can
find the newforms using modular symbols for
,
which has half the dimension of
; but to find the
exact period lattice requires working in
.) The
argument is similar to the one given above, using
.
**

**
**

First assume that .

If the type of is the same as that of (for example, this must be the case if all the have the same type, which will hold whenever all the isogenies between the have odd degree) then from we deduce as before that exactly, and , hence . So in this case we have that , though there might be some ambiguity in which curve is optimal if for more than one value of .

Assume next that has type but has type . Now . The usual argument now gives . Hence either and , or and . To see if the latter case could occur, we look for classes in which and has type , while for some we also have and of type . This occurs 28 times for , but for 15 of these the level is odd, so we know that must be odd. The remaining 13 cases are

we have been able to eliminate these by carrying out the extra computations necessary as in the proof of Theorem 5.2. We note that in all of these 13 cases, the isogeny class consists of two curves, of type 1 and of type 2, with , so that rather than has minimal Faltings height.

Next suppose that has type but has type . Now . The usual argument now gives , which is impossible; so this case cannot occur.

Finally we consider the cases where . There are only three of these for : namely, , and , where . In each case the all have the same type (they are linked via -isogenies) and the usual argument shows that . But none of these levels is divisible by , so in each case.

**
**