Manin constants and optimal curves: conductors 60000-300000
-----------------------------------------------------------

For all conductors (levels) N up to 60000 we have computed the full
modular symbol space for Gamma_0(N) and hence know both that the first
curve in the class is the Gamma_0(N)-optimal one and that the Manin
constant c is equal to 1, as stated in my Appendix to "The Manin
Constant" by Amod Agashe, Ken Ribet and William Stein [Pure and
Applied Mathematics Quarterly, Vol. 2 no.2 (2006), pp. 617-636.]

Here we report on what is known for larger conductors, currently
covering the range 60000-300000.

# isogeny classes:                         1051602
# isogeny classes with only one curve:      726443
# isogeny classes with more than one curve: 325159

Of the latter, c=1 known for:               325125+34=325159
               optimality known for:        151293+34=151327
               optimality not known for:              173832
[141444,29583,2537,124,144 classes have 2,3,4,5,6 possible candidates]

For all 726443 classes with only one curve, obviously that curve is
optimal, and we have shown that c=1.

For all classes with more than one curve, we have proved that the
optimal curve has c=1, but cannot yet say unconditionally which is the
optimal curve in all cases (only in at least 83% of cases).

This is because in most cases for N>60000 we have only computed the +1
modular symbols and therefore can only compare real periods, as
described in the Appendix to Agashe-Ribet-Stein for the range
60000-130000; and this is not always sufficient to single out the
optimal curve in the class.

The theoretical results on the Manin constant c which we have used
are: that c is an integer, that c=2 is impossible when N is odd, and
that c=3 is impossible unless N is a multiple of 3.

-----------------------------------------------------------

In detail,  for the 325159 classes with more than one curve:

In 151327 classes the optimal curve is provably the first one listed,
and c=1.

In 173832 classes the optimal curve is one of up to 6 possible curves
in the class, but c=1 in any case.

e.g. 130050em: class has 8 curves
               types: 1 2 1 2 2 2 2 2
               a_j:   1 2 1 2 4 2 2 4
 -either the optimal curve has type 1 and is #1 or #3 with c=1 [c*aj=1]
 -or the optimal curve has type 2 and is #2, #4, #6, or #7 with c=1 [c*aj=2]

Note that for 34 classes, there would be a possibility that c=2 and
that the optimal curve is the second one listed, if we relied only on
the information given by modular symbols with sign +1.  These all have
the same form: 2 curves in the class with types 1,2 and aj=1,1 so
either E1 is optimal with c=1 or E2 is optimal with c=2.  The minimal
period lattices of E1, E2 have the form [2x,x+yi], [2x,2yi] with x,y
positive real, and from the +1 modular symbols we can only say that
the projection of the period lattice of the normalised newform onto
the real line is generated by x; so the optimal curve might
conceivably have lattice [x,yi], implying that E2 is optimal with c=2.

These classes are: (13 listed in the appendix to the paper cited),
                   133972a, 144464a, 149012a, 150608j, 164852a,
                   169808a, 171412c, 184916a, 188372a, 211664a,
                   217172b, 219088b, 220916b, 236212b, 240116a,
                   250064a, 256052a, 260116a, 280916a, 285172a, 291664a.

In all of the above cases I have computed the full modular symbol
space to eliminate the second possibility.  Hence, in this range all
optimal curves are proved to have c=1, and in at least 83% of cases,
the optimal curve is known (and is the first curve in the class).

I have also computed the modular degrees of all curves (not just the
optimal ones) using Mark Watkins's sympow program, which confirms
optimality of the first curve in each class conditional on Stevens's
conjecture that the Gamma_1(N)-optimal curve is the one with minimal
Faltings Height (i.e. the one whose period lattice is a sublattice of
all the others).

The files optimality.06, ..., optimality.29 contain one line for each
class of size at least 2, with the following format:

209990bb: c=1; 3 possible optimal curves: E1 E2 E3
209990bc: c=1; optimal curve is E1
209990be: c=1; optimal curve is E1
184916a: Neither c nor the optimal curve are uniquely determined.  Possibilities for (c,j0) are: (1,1) (2,2)

The last of these is one of the 33 cases which were ambiguous before
computing the full modular symbol space, but have now been settled; so
it should now read:

184916a: c=1; optimal curve is E1