m=1 p=1000033 k=2: 8 cubics
all cubics smooth mod p: True
each cubic through exactly its 9 base vertices; at each vertex 6 cubics with 2 tangent lines, 3 cubics from 3 pencils per line, none a configuration line: True
all 9 intersection points of every pair found over F_p: True (0s)
  pair type cross vertex mults (1, 1, 1, 2, 2, 2) off-vertex mults () : 24 pairs
  pair type same vertex mults (1, 1, 1, 1, 1, 1, 1, 1, 1) off-vertex mults () : 4 pairs
off-vertex intersection points: 0  (number of cubics through it, pencils): {}
quadruple points per cubic (value: number of cubics): {0: 8}
expected number of quadruple points 9k(m^2-1) = 0
Z: t=8 monomials 45 conditions 60 rank 45 h1 15 dim I_t 0  (0s)
Z: t=9 monomials 55 conditions 60 rank 52 h1 8 dim I_t 3  (0s)
Z: t=10 monomials 66 conditions 60 rank 57 h1 3 dim I_t 9  (0s)
Z: t=11 monomials 78 conditions 60 rank 60 h1 0 dim I_t 18  (0s)
Z: t=12 monomials 91 conditions 60 rank 60 h1 0 dim I_t 31  (0s)
3Z: t=23 monomials 300 conditions 396 rank 300 dim I(3Z)_t 0  (0s)
3Z: t=24 monomials 325 conditions 396 rank 324 dim I(3Z)_t 1  (0s)
  product of the cubics satisfies all 3Z conditions: True
