Revision

**f69d0e06f67b9ac6f57e7d8e6ba3b3d69e650352**authored by Maarten Derickx on**02 November 2020, 22:43:35 UTC**, committed by Maarten Derickx on**02 November 2020, 22:43:35 UTC****1 parent**0460952

X_1_m_n.m

```
function X_1_m_n(m,n,base_ring : equation_directory:="models_X1_m_n", zeta_m:=0)
//Input: m,n - integers such that m divides n
// base_ring - a ring
// equation_directory - directory with files X1_m_n.txt containing models
// zeta_m - a primitive mth root of unity in the base_ring (if unspecified one will be chosen)
//Output: C - a curve
//Returns an algebraic model C of the modular curve X_1(m,n) as a curve over base_ring
assert IsDivisibleBy(n,m);
if m gt 2 then
if zeta_m ne 0 then
assert zeta_m^m eq 1 and &and[zeta_m^e ne 1: e in Divisors(m)| e ne m];
else
try
zeta_m := RootOfUnity(m,base_ring);
catch e
printf "Specified base ring %o does not contain a %oth root of unity", base_ring, m;
assert false;
end try;
end if;
z:=zeta_m; i:=zeta_m;
end if;
n_str := IntegerToString(n);
m_str := IntegerToString(m);
file_name := equation_directory cat "/X1_" cat m_str cat "_" cat n_str cat ".txt";
data := Read(file_name);
data := Split(data);
//example contents of the file X1_2_10.txt
//X := v^2 + (u^2 - 1)*v - 1;
//q := 1/u;
//t := -4*u/(u^2*v + u^2 - v + 3);
//E:=[0,t^2-2*q*t-2,0,-(t^2-1)*(q*t+1)^2,0];
//P:=[(t+1)*(q*t+1),t*(q*t+1)*(t+1)];
//Q:=[0,0];
A<u,v> := AffineSpace(base_ring,2);
for line in data do
val := Split(Split(line,"=")[2],";")[1];
if line[1] eq "X" then X := eval(val); end if;
if line[1] eq "q" then q := eval(val); end if;
if line[1] eq "r" then r := eval(val); end if;
if line[1] eq "s" then s := eval(val); end if;
if line[1] eq "t" then t := eval(val); end if;
if line[1] eq "E" then E := eval(val); end if;
if line[1] eq "P" then P := eval(val); end if;
if line[1] eq "Q" then Q := eval(val); end if;
end for;
C := Curve(A,X);
return ProjectiveClosure(C),E,P,Q;
end function;
```

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