Inequality in Theorem 32 for q geq 5.txt

fixedpointsALinvsmall:=function(m, n)

if n eq 3 then
q:=Factorization(Numerator(m/3));
nu_3:=2;
for d in [1 .. #q] do
if q[d][1] eq 2 then
if q[d][2] gt 2 then
nu_3:=0;
else
end if;
else
nu_3:=nu_3*(1+LegendreSymbol(-n,q[d][1]));
end if;
end for;
return nu_3;
else
if n eq 1 then
else
if n eq 2 then
q:=PrimeDivisors(Numerator(m/n));
nu_2factorminus1:=1;
nu_2factorminus2:=1;
for d in [1 .. #q] do
nu_2factorminus1:=nu_2factorminus1*(1+LegendreSymbol(-1,q[d]));
nu_2factorminus2:=nu_2factorminus2*(1+LegendreSymbol(-2,q[d]));
end for;
return nu_2factorminus1+nu_2factorminus2;
else
if n eq 4 then
q:=PrimeDivisors(Numerator(m/n));
l:=Divisors(Numerator(m/n));
nu_4factorminus1:=1; nu_4sumeuler:=0;
for d in [1..#q] do
nu_4factorminus1:=nu_4factorminus1*(1+LegendreSymbol(-1,q[d]));
end for;
for dd in [1..#l] do
ss:=Numerator(m/(n*l[dd]));
ssh:=GCD(l[dd],ss);
nu_4sumeuler:=nu_4sumeuler+EulerPhi(ssh);
end for;
return nu_4factorminus1+nu_4sumeuler;
end if;
end if;
end if;
end if;

end function;



fixedpointsALinvbig:=function(m, n)

PD:=PrimeDivisors(Numerator(m/n)); D:=Divisors(Numerator(m/n));

n_mod2:=n mod 2; n_mod4:=n mod 4;


if n_mod2 eq 1 then
if n_mod4 eq 3 then
if 2 in PD then
if 8 in D then
nu_nodd3mod4_8:=2*(ClassNumber(-n)+ClassNumber(-4*n))*(1+KroneckerSymbol(-n,2));
if #PD eq 1 then
else
for p1 in PD do
p1_mod2:=p1 mod 2;
if p1_mod2 eq 1 then
nu_nodd3mod4_8:=nu_nodd3mod4_8*(1+LegendreSymbol(-n,p1));
end if;
end for;
end if;
return nu_nodd3mod4_8;
else
if 4 in D then
nu_nodd3mod4_4:=
(2*ClassNumber(-4*n)+2*(1+KroneckerSymbol(-n,2))*ClassNumber(-n));
if #PD eq 1 then
return nu_nodd3mod4_4;
else
for p2 in PD do
p2_mod2:=p2 mod 2;
if p2_mod2 eq 1 then
nu_nodd3mod4_4:=nu_nodd3mod4_4*(1+LegendreSymbol(-n,p2));
end if;
end for;
return nu_nodd3mod4_4;
end if;
else
nu_nodd3mod4_2:=ClassNumber(-4*n)+3*ClassNumber(-n);
if #PD eq 1 then
return nu_nodd3mod4_2;
else
for p3 in PD do
p3_mod2:=p3 mod 2;
if p3_mod2 eq 1 then
nu_nodd3mod4_2:=nu_nodd3mod4_2*(1+LegendreSymbol(-n,p3));
end if;
end for;
return nu_nodd3mod4_2;
end if;
end if;
end if;
else
if #PD eq 0 then
nu_nodd3mod4_no2:=ClassNumber(-n)+ClassNumber(-4*n);
return nu_nodd3mod4_no2;
else
nu_nodd3mod4_no2:=ClassNumber(-n)+ClassNumber(-4*n);
for j in [1..#PD] do
nu_nodd3mod4_no2:=nu_nodd3mod4_no2*(1+LegendreSymbol(-n,PD[j]));
end for;
return nu_nodd3mod4_no2;
end if;
end if;
else
if #PD eq 0 then
nu_nodd1mod4:=ClassNumber(-4*n);
return nu_nodd1mod4;
else
if 2 in PD then
if 4 in D then
return 0;
else
nu_nodd1mod4_2:=ClassNumber(-4*n);
PD2:=PrimeDivisors(Numerator(m/(2*n)));
for k in [1..#PD2] do
nu_nodd1mod4_2:=nu_nodd1mod4_2*(1+LegendreSymbol(-n,PD2[k]));
end for;
return nu_nodd1mod4_2;
end if;
else
nu_nodd1mod4_no2:=ClassNumber(-4*n);
for i in [1 .. #PD] do
nu_nodd1mod4_no2:=nu_nodd1mod4_no2*(1+LegendreSymbol(-n,PD[i]));
end for;
return nu_nodd1mod4_no2;
end if;
end if;
end if;
else
nu_neven:=ClassNumber(-4*n);
if #PD eq 0 then
return nu_neven;
else
for u in [1 .. #PD] do
nu_neven:=nu_neven*(1+LegendreSymbol(-n,PD[u]));
end for;
return nu_neven;
end if;
end if;
end function;


generexoN:=function(b)

m:=PrimeDivisors(b);
l:=Divisors(b);
factor_b:=Factorization(b);
psiEulerindex:=b;
order4elliptic:=1;
order3elliptic:=1;
cusps:=0;

for x in [1 .. #m] do
psiEulerindex:=psiEulerindex*(1+1/m[x]);
end for;

for y in [1..#m] do
if factor_b[y][1] eq 2 then
order3elliptic:=0;
if factor_b[y][2] gt 1 then
order4elliptic:=0;
end if;
else
if factor_b[y][1] eq 3 then
if factor_b[y][2] gt 1 then
order3elliptic:=0;
order4elliptic:=order4elliptic*(1+LegendreSymbol(-1,factor_b[y][1]));
else
order3elliptic:=order3elliptic*(1+LegendreSymbol(-3,factor_b[y][1]));
order4elliptic:=order4elliptic*(1+LegendreSymbol(-1,factor_b[y][1]));
end if;
else
order4elliptic:=order4elliptic*(1+LegendreSymbol(-1,factor_b[y][1]));
order3elliptic:=order3elliptic*(1+LegendreSymbol(-3,factor_b[y][1]));
end if;
end if;
end for;

for a in [1 .. #l] do
n1:=Numerator(b/l[a]);
t1:=GCD(l[a],n1);
cusps:=cusps+EulerPhi(t1);
end for;

genus:=1+(psiEulerindex/12)-(order4elliptic/4)-(order3elliptic/3)-(cusps/2);
return genus;
end function;


genereX0NQuotientWN:=function(N,WN,t);

FixedpointsALinvolutions:=[* *]; vv:=0; L:=Divisors(N);


for i in [1..#WN] do
u:=GCD(WN[i],Numerator(N/WN[i]));
if WN[i] in L then
else
vv:=1;
end if;

if u eq 1 then
if WN[i] eq 1 then
else
if WN[i] gt 4 then
nu_Ddi:=fixedpointsALinvbig(N,WN[i]);
FixedpointsALinvolutions:=Append(FixedpointsALinvolutions,nu_Ddi);
else
nu_Ddi:=fixedpointsALinvsmall(N,WN[i]);
FixedpointsALinvolutions:=Append(FixedpointsALinvolutions,nu_Ddi);
end if;
end if;
else
vv:=1;
end if;

end for;

CountAllFixedPointsALinvolutions:=0;

for u in FixedpointsALinvolutions do
CountAllFixedPointsALinvolutions:=CountAllFixedPointsALinvolutions+u;
end for;
if vv eq 0 then
genusxoN:=generexoN(N);
genusxoNQuotient:=1+t^(-1)*(genusxoN-1-(CountAllFixedPointsALinvolutions/2));
else
genusxoNQuotient:=-1;
end if;
return genusxoNQuotient;

end function;


JacobianDecompositionQuotientX0NWN:=function(n,WN,length,t); N:=n;
AtkinLehnerfix:=WN;
Involutions:=#AtkinLehnerfix;
L:=[**];
F:=[**];
Level:=[* *];
ALaction:=[**];

Pd:=PrimeDivisors(N); mt:=[**];

for prime in Pd do
op:=Valuation(N,prime);
primepower:=Gcd(prime^(op), N);

mt:=Append(mt, primepower);
end for;


Nd:=Divisors(N); countergenus:=0;

for j in Nd do

MS:=NewformDecomposition(CuspidalSubspace(ModularSymbols(j,2,1)));

m:=#MS;

M:=PrimeDivisors(j);

Nr:=Numerator(N/j);

divi:=GCD(j,Nr);

jj:=Numerator(j/divi);

Mm:=PrimeDivisors(jj);

Nn:=Divisors(jj);

mm:=#Mm;

mn:=#Nn;

D:=Factorization(jj);

for i in [1..m] do

f:=Eigenform(MS[i],length);

f2:=MS[i];

K:=Parent(Coefficient(f,3)); d:=Dimension(MS[i]);

X:=IdentityMatrix(Rationals(), d);

u:=0;
for jo in [1..Involutions] do
dd:=GCD(j,AtkinLehnerfix[jo]);

if dd eq AtkinLehnerfix[jo] then

Y:=AtkinLehner(MS[i],dd);

if Y eq X then

else

u:=1;

end if;

else

if dd eq 1 then

else

end if;

end if;

end for;

if u eq 0 then
ALactionf:=[**];
for i in [1..#Pd] do
if GCD(mt[i],j) eq mt[i] then

ALactionf:=Append(ALactionf, [*AtkinLehner(f2,mt[i]),mt[i]*]);


end if;
ALaction:=Append(ALaction,[*f,j,ALactionf*]);
end for;


L:=Append(L,f);


F:=Append(F,K);

uu:=#Basis(K);

countergenus:=countergenus+uu;

Level:=Append(Level,j);
else
end if;
end for;

end for;
genuscorrect:=genereX0NQuotientWN(N,WN,t);

if genuscorrect eq countergenus then

CorrectJacobian:=11111111111111;

else

CorrectJacobian:=0000000000000;

end if;
M:=[*CorrectJacobian,L,F,Level, ALaction*];

return M;

end function;

FpnpointsforQuotientcurveX0NWN:=function(N,prime,JacDecomp,FieldDefinition,bound);

L:=JacDecomp;
p:=prime;
F:=FieldDefinition;
felm:=# F;
bod:=bound;


C:=ComplexField(100); R<x>:=PolynomialRing(C); pj:=0*x+1; Roo:=[**];
for j in [1 .. felm] do

if Degree(F[j]) eq 1 then

cc:=Roots(x^2-Coefficient(L[j],p)*x+p,C);

Roo:=Append(Roo,cc);

pj:=pj*(x^2-Coefficient(L[j],p)*x+p);

else

dd:=Degree(F[j]);

u:=Roots(DefiningPolynomial(F[j]),C); uu:= # u;

for m in [1 .. uu] do

f := hom< F[j] -> C | u[m][1]>;

cc2:=Roots(x^2-f(Coefficient(L[j],p))*x+p,C);

Roo:=Append(Roo,cc2);

pj:=pj*(x^2-f(Coefficient(L[j],p))*x+p);

end for;

end if;

end for;
pjdegree:=Degree(pj);

PR:=[* *];

d2:=Degree(pj);

long:= # Roo;





for nn in [1 .. bod] do s:=0;

for i in [1 .. long] do

for j in [1..2] do

if Roo[i][j][2] gt 0 then

s:=s+(Roo[i][j][2])*(Roo[i][j][1])^(nn) ;

else

s:=s;

end if;

end for;
end for;
a:=Round(1+p^(nn)-s);
PR:=Append(PR,a);
end for;

return PR;
end function;


MapdegreedtoEC:=function(prime,degree,bound,FpNpointsModularCurveList)

p:=prime;
// a3:=apCoefficientEC; 
bod:=bound; deg:=degree;

PR2:=[* *];

C:=ComplexField(100); R<x>:=PolynomialRing(C);
//cearrels:=Roots(x^2-a3*x+p,C);



for i in [1..bod] do



b:=deg*(p^i+1);

PR2:=Append(PR2,b); end for;

el:=#FpNpointsModularCurveList; tt:=Min(el,bod);

NoDegreeMaptosuchEC:=[**];

for k in [1..tt] do

difference:=(FpNpointsModularCurveList[k])-(PR2[k]);

Rr:=RealField(10); difference:=Rr!difference;

case Sign(difference):
when 1:
NoDegreeMaptosuchEC:=Append(NoDegreeMaptosuchEC,[*difference,p^k*]);

end case;




end for;

return NoDegreeMaptosuchEC;

end function;

Squarefreepart:=function(Integer)

n:=Integer;
Pn:=#PrimeFactors(n);

for p in PrimeFactors(n) do
if p^2 in Divisors(n) then
n:=Truncate(n/(p^2));
end if;
end for;

return n;

end function;


A1:=[* *];
A2:=[* *];
n:=2;
for q in PrimesInInterval(5,100) do
for M in [2..144] do
if IsSquarefree(q*M) eq true then
if #PrimeFactors(M) le 4 then

//if (SumOfDivisors(M)/12 +2^(#PrimeFactors(M))) le (2*n*(q^2+1)) then
for d in Divisors(M) do
if d gt 1 then 
TN1:=[*d,1*];
t1:=#TN1;
A11:=genereX0NQuotientWN(q*M,TN1,t1);
B11:=genereX0NQuotientWN(M,TN1,t1);
if A11 ge 2 then
if (A11-1) ge (n*(A11-1-(2*B11))) then
A1:=Append(A1,[q,M,d]);
end if;
end if;
end if;
end for;
//end if;

if #PrimeFactors(M) in [2..4] then
//if (SumOfDivisors(M)/12 +2^(#PrimeFactors(M))) le (4*n*(q^2+1)) then
for d1 in Divisors(M) do
if d1 gt 1 then
for d2 in Divisors(M) do
if d2 gt 1 then
if d1 ne d2 then
TN2:=[*d1,d2,Squarefreepart(d1*d2),1*];
t2:=#TN2;
A12:=genereX0NQuotientWN(q*M,TN2,t2);
B12:=genereX0NQuotientWN(M,TN2,t2);
if A12 ge 2 then
if (A12-1) ge (n*(A12-1-(2*B12))) then
A1:=Append(A1,[q,M,d1,d2]);
end if;
end if;
end if;
end if;
end for;
end if;
end for;
//end if;
end if;

if #PrimeFactors(M) in [3..4] then
//if (SumOfDivisors(M)/12 +2^(#PrimeFactors(M))) le (8*n*(q^2+1)) then
for d1 in Divisors(M) do
if d1 gt 1 then
for d2 in Divisors(M) do
if d2 gt 1 then
if d1 ne d2 then
for d3 in Divisors(M) do
if d3 gt 1 then
if d3 notin [d1,d2,Squarefreepart(d1*d2)] then
TN3:=[d1,d2,d3,Squarefreepart(d1*d2), Squarefreepart(d1*d3), Squarefreepart(d2*d3),Squarefreepart(d1*d2*d3),1];
t3:=#TN3;
A13:=genereX0NQuotientWN(q*M,TN3,t3);
B13:=genereX0NQuotientWN(M,TN3,t3);
if A13 ge 2 then
if (A13-1) ge (n*(A13-1-(2*B13))) then
A1:=Append(A1,[q,M,d1,d2,d3]);
end if;
end if;
end if;
end if;
end for;
end if;
end if;
end for;
end if;
end for;
//end if;
end if;

if #PrimeFactors(M) eq 4 then
//if (SumOfDivisors(M)/12 +2^(#PrimeFactors(M))) le (16*n*(q^2+1)) then
TN4:=Divisors(M);
t4:=#TN4;
A14:=genereX0NQuotientWN(q*M,TN4,t4);
B14:=genereX0NQuotientWN(M,TN4,t4);
if A14 ge 2 then
if (A14-1) ge (n*(A14-1-(2*B14))) then
A1:=Append(A1,[q,M,PrimeFactors(M)[1], PrimeFactors(M)[2], PrimeFactors(M)[3], PrimeFactors(M)[4]]);
end if;
end if;
//end if;
end if;

end if;
end if;
end for;
end for;
A1;