\\======================================================================
\\ Ellenberg. q = conductor character attached to K/Q.

bound1(p,q)=3*30*(400/399)^3*exp(2*Pi)/p^3*q^2;

E1(p,q)=
{local(Sigma);
Sigma=q^2/2/Pi;
16/3*q^2*Pi^2*log(p)/exp(p^2/4/Pi/Sigma/log(p))
}

E2(p,q)=8/9*Pi^3*zeta(7/2)^2*q^4/p^3*log(p)^2

E3(p,q)=8*zeta(3/2)^2*Pi^2*q^2/p*log(p)*exp(-p^2/q^2/log(p^2));

E4(p,q)=
{local(Sigma);
Sigma=q^2/2/Pi;
16*Pi^3*(2/Pi*eulerphi(q)*6*log(p)^2/p^2+Sigma/2*log(p^2)/p*(zeta(3/2)^2-sum(i=1,p^2,sigma(i,0)/sqrt(i^3))))}

DJ(p)=1280*Pi^2*log(p)^2/p^2+1536/3*Pi^2*log(p)/p*(zeta(3/2)^2-sum(i=1,p,sigma(i,0)/sqrt(i^3)))

E5(p,q,bd)=
{local(Sigma,A,B,C);
Sigma=sigma(q,0)/2/Pi;
A=2/Pi*sigma(q,0);
B=Sigma/6*p^2*log(p^2);
16*Pi^3*sum(i=1,bd,min(A*log(p^2*i)/p^2/i,B*sigma(p^2*i,0)/p^3/sqrt(i^3)))
}

F(p,q)=
{local(Sigma);
Sigma=q^2/2/Pi;
4*Pi*exp(-Pi/Sigma/p^2/log(p))-E4(p,q)-E3(p,q)-E2(p,q)-E1(p,q)-bound1(p,q)}

F2(p,q,m)=2*sqrt(3)*sqrt(m)*sigma(m,0)/(1-exp(-2*Pi/q/sqrt(p)))*(4*Pi+16*zeta(3/2)^2*Pi^2/sqrt(p^3))

Bd2(p,q)=1/(p^2-1)*F2(p,q,p)

Bd3(p,q)=p/(p^2-1)*F2(p,q,1)

\\======================================================================
\\ This is the script to compute the bound. q = conductor K/Q. The first p where the output is posiive is the right answer.

EllenbergBound(p,q)=F(p,q)-1/(p^2-1)*F2(p,q,p)-p/(p^2-1)*F2(p,q,1)