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