Example: A piece of hyperelliptic 7-torsion
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))
We obtain a polynomial with Galois group PGL2(F7) which
ramifies only at 7 and at 13.
Nicolas Mascot p-adic computation of Galois representations