{
  "domain": "V = Sel_11(E/Q) = E(Q)/11 (Sha[11] = 0 certified; torsion trivial), basis P = (0,0), Q = (1,0) - the Mordell-Weil generators (P5 v1.2; RUN-011)",
  "codomain": "W (x) I/I^2 with W = (+)_l E(F_l) (x) F_11 = (+)_l H^1_f(Q_l, E[11]) (Kummer), basis e_l a chosen generator; E(F_397), E(F_991) cyclic of orders 374, 1045 with v_11 = 1 (RUN-067)",
  "base_ring": "A = F_11[G], G = G_397 x G_991 = C_11 x C_11 = the 11-part of Gal(Q(zeta_397 zeta_991)/Q); I its augmentation ideal; X_l = gamma_l - 1 for CHOSEN generators gamma_l",
  "cohomological_origin": "the restriction map H^1_f(Q, E[11]) -> (+)_l H^1_f(Q_l, E[11]) - canonical, no choice - composed with the isomorphism E(Q_l)(x)F_11 = E(F_l)(x)F_11 (good reduction) and then tensored with X_l",
  "connecting_morphism": "none: B_N is not the boundary of any exact triangle; the tensor with X_l is the canonical identification I_l/I_l^2 = G_l (x) F_11 evaluated on a chosen generator",
  "bases": "P, Q (Mordell-Weil, canonical up to GL_2(Z)); e_397, e_991 (generators of cyclic groups, a unit choice each); gamma_397, gamma_991 (generators of C_11, a unit choice each)",
  "verdict_hand_built_or_derived": "hand-built: the canonical localization matrix M_loc dressed with augmentation generators; not a Bockstein morphism induced from a Selmer / Euler-system complex",
  "kurihara_condition_a_l_minus_l_minus_1_mod_11": {
    "397": 0,
    "991": 0
  },
  "v_11_of_counts": {
    "397": 1,
    "991": 1
  },
  "cyclic": {
    "397": true,
    "991": true
  },
  "M_loc": [
    [
      1,
      2
    ],
    [
      1,
      4
    ]
  ],
  "det_M_loc_mod_11": 2,
  "det_B_N": "2 * (e_397 ^ e_991) (x) X_397 X_991",
  "matches_RUN_069": true,
  "agrees": true
}