{
  "selmer_rank_2": true,
  "independence_of_P_Q_in_Sel": "from det(M_loc) = 2 != 0: the images of P, Q in (+) E(F_l)(x)F_11 are independent, hence P, Q are independent in Sel",
  "from_rank_alone": false,
  "kato_source_rank": "nu(n) = 2 = ord(delta-tilde) = rank E(Q) (Kim, with Sha[11] = 0)",
  "local_conditions": {
    "397, 991": "H^1_f = E(Q_l)(x)F_11 = E(F_l)(x)F_11, loc_n an isomorphism onto their sum (Kim)",
    "11": "E(Q_11)[11] = 0; ordinary, a_11 = -4"
  },
  "agrees": true
}