Magma V2.30-1
q = 3: possible values of P(1): [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 34, 35, 36, 42, 49 ]
q = 2: possible values of P(1): [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 25 ]
CHECK no abelian surface over F_3 has 26 or 52 points; none over F_2 has a multiple of 39 points: ok
CURVE Hyperelliptic Curve defined by y^2 + (x^3 + x^2 + 1)*y = x^2 + x over Rational Field
J(Q)_tors = Z/19
isomorphic to y^2 + (x^3 + x + 1) y = x^5 + x^4 (LMFDB 169.a.169.1)
characteristic polynomial of Frobenius at 2: x^4 + 3*x^3 + 5*x^2 + 6*x + 4
CHECK P_2 has no root mod 13, so J[13]^ss has no one-dimensional constituent: ok
#J(F_2) = 19
CHECK #J(F_2) is prime to 3, so J[3]^ss has no trivial constituent: ok
ALL_CHECKS_PASSED