J = smoothplanepicinit(x^4+2*y^4+x^3-3*x*y-2,29,3)
W = picrand(J)
picmember(J,W)
piciszero(J,W)
W2 = picrand(J);
piceq(J,W,W2)
picadd(J,W,W2)

factor(piccard(J))
W = picrandtors(J,13);
picmember(J,W)
piciszero(J,picmul(J,W,13))
piciszero(J,W)
picistorsion(J,W,13)

P = pictorspairinginit(J,13);
X = picrand(J);
pictorspairing(J,P,W,X)
pictorspairing(J,P,picmul(J,W,2),X)

FW = picfrob(J,W);
pictorspairing(J,P,FW,X)
piceq(J,picmul(J,W,9),picfrob(J,W))

J2 = picsetprec(J,21); \\ Now mod 29^e, e=21
Y = picrand(J2)
picmul(J2,Y,-3)
picmember(J2,W)
picmemberval(J2,W)
picmemberval(J2,Y)

W2 = piclifttors(J2,W,13);
picmember(J2,W2)
picistorsion(J2,W2,13)
piciszero(J2,W2)
piceq(J2,picmul(J2,W2,9),picfrob(J2,W2))

f = x^3*y+y^3+x;
P = [1,0,0]; \\ Points on C
Q = [0,1,0]; \\ Needed to construct J -> A1
l = 2; \\ Look at J[2]
p = 5; e = 60; \\ Work mod 5^60
R = smoothplanegalrep(f,l,p,e,[[P],[Q]])
fa = factor(R[1])
Mat(apply(polredabs,fa[,1]))

h = x^3+x+1; \\ C : y^2+h(x)*y = f(x)
f = x^5+x^4; \\ Good reduction away from 13
P = [-1,0]; \\ Points on C
Q= [0,0]; \\ Needed to construct J -> A1
p = 17; e = 30; \\ Work mod 17^30
l = 7; \\ Look at piece of J[7]
chi = x^2-x-2;  \\ Where Frob17 acts like this
R = hyperellgalrep([f,h],l,p,e,[P,Q],chi)
PR = projgalrep(R);
F = polredabs(PR[1])
polgalois(F)
factor(nfdisc(F))

S = mfinit([16,2,0],1);
f = mfeigenbasis(S[1])[1];
R = mfgalrep(f,[5,[[2,2]]],[30,50],5)
factor(projgalrep(R)[1])

f = mfDelta();
R = mfgalrep(f,17,100,200)
F = polredbest(projgalrep(R)[1])
factor(nfdisc(F))