{
  "coefficient_field": "F_11",
  "curve": "y^2+y=x^3+x^2-2x",
  "short_model": "Y^2=X^3-3024X+46224",
  "coordinate_change": {
    "X": "36*x+12",
    "Y": "216*y+108"
  },
  "minimal_invariants": {
    "b2": 4,
    "b4": -4,
    "b6": 1,
    "b8": -3,
    "c4": 112,
    "discriminant": 389,
    "discriminant_is_prime": true
  },
  "minimal_model_point_counts": {
    "2": 5,
    "5": 9,
    "11": 16
  },
  "residual_irreducibility_witness": {
    "frobenius_prime": 2,
    "a_2": -2,
    "characteristic_polynomial_mod_11": [
      1,
      2,
      2
    ],
    "discriminant_mod_11": 7,
    "square_residues_mod_11": [
      0,
      1,
      3,
      4,
      5,
      9
    ],
    "roots_mod_11": [],
    "note": "Transvection and subgroup arguments are proved in the manuscript, not computed here."
  },
  "good_at_11": {
    "group_order": 16,
    "a_11": -4
  },
  "local_results": [
    {
      "ell": 397,
      "group_order": 374,
      "a_ell": 24,
      "order_of_P": 374,
      "Q_discrete_log_base_P": 244,
      "prime_to_11_multiplier": 34,
      "multiplied_P": [
        281,
        236
      ],
      "multiplied_Q": [
        11,
        334
      ],
      "order_of_multiplied_P": 11,
      "Q_norm_coordinate": 2,
      "independent_counts_agree": true,
      "P_orbit_equals_all_points": true
    },
    {
      "ell": 991,
      "group_order": 1045,
      "a_ell": -53,
      "order_of_P": 1045,
      "Q_discrete_log_base_P": 356,
      "prime_to_11_multiplier": 95,
      "multiplied_P": [
        39,
        97
      ],
      "multiplied_Q": [
        865,
        243
      ],
      "order_of_multiplied_P": 11,
      "Q_norm_coordinate": 4,
      "independent_counts_agree": true,
      "P_orbit_equals_all_points": true
    }
  ],
  "matrix": [
    [
      1,
      2
    ],
    [
      1,
      4
    ]
  ],
  "determinant_mod_11": 2,
  "source_images_count": 121,
  "mixed_polynomial_coefficients": {
    "X^2": 0,
    "X*Y": 2,
    "Y^2": 0
  },
  "simultaneous_reduction_lattice": {
    "basis": [
      [
        5742,
        -22
      ],
      [
        -254980,
        1045
      ]
    ],
    "congruence_residues": [
      [
        0,
        0
      ],
      [
        0,
        0
      ]
    ],
    "index": 390830,
    "target_order": 390830
  },
  "falsifying_witnesses": {
    "nonzero_mod_121_but_zero_mod_11": {
      "integer": 11,
      "mod_121": 11,
      "mod_11": 0
    },
    "duplicate_row_determinant": 0,
    "abstract_selmer_countermodel": {
      "matrix": [
        [
          1,
          2,
          0,
          0
        ],
        [
          1,
          4,
          0,
          0
        ]
      ],
      "source_dimension": 4,
      "image_dimension": 2,
      "kernel_dimension": 2,
      "elliptic_curve_realization_claimed": false
    },
    "same_special_fiber_different_deformation": {
      "fiber_differential": [
        [
          0,
          0
        ],
        [
          0,
          0
        ]
      ],
      "lift_0": [
        [
          "0",
          "0"
        ],
        [
          "0",
          "0"
        ]
      ],
      "lift_1": [
        [
          "X",
          "2*X"
        ],
        [
          "Y",
          "4*Y"
        ]
      ],
      "first_bockstein_zero": true,
      "second_bockstein_zero": false,
      "determinant_in_associated_graded_degree_2": "2*X*Y",
      "determinant_inside_square_zero_ring_itself": "0"
    }
  },
  "not_computed": [
    "canonical modular symbols or Kurihara number",
    "full Selmer group",
    "Sha",
    "global Selmer-complex regulator",
    "complex L-value or height",
    "formal proof assistant verification"
  ]
}