{
  "source": "LR Assumption B1.1 (throughout Part B), C1.1's import of Section B3.1, and section B2's Kato-family assumptions",
  "partners": {
    "389.a1": {
      "weierstrass": {
        "c4": 112,
        "c6": -856,
        "discriminant": 389,
        "identity_1728_disc_eq_c4_cubed_minus_c6_squared": true
      },
      "discriminant_factorisation": {
        "389": 1
      },
      "hypotheses": {
        "ordinary at p (a_p not divisible by p)": true,
        "unit root mod p": 7,
        "prime-to-p level (p does not divide N)": true,
        "Hecke polynomial X^2 - a_p X + p has distinct roots (disc nonzero)": true,
        "p-distinguished (the two roots differ mod p)": true,
        "noble (prime-to-p level + distinct Hecke roots), Assumption B1.1": true,
        "non-critical slope (ordinary means slope 0 < r_1 + 1)": true,
        "Hida (ordinary), as section B2 restricts to": true,
        "newform of its level, as section B2 assumes": true,
        "weight 2, trivial nebentypus": true,
        "minimal model (Delta squarefree, and 1728 Delta = c_4^3 - c_6^2)": true,
        "semistable (no additive reduction: every prime dividing Delta fails to divide c_4)": true,
        "residually irreducible at p = 11 (semistable + Mazur: a rational l-isogeny forces l <= 7)": true
      },
      "hecke_polynomial": {
        "discriminant": -28,
        "discriminant_mod_p": 5
      },
      "multiplicative_at_each_bad_prime": {
        "389": true
      },
      "semistable": true,
      "discriminant_squarefree": true,
      "minimal_model_guard": true,
      "all_hold": true
    },
    "19.a1": {
      "weierstrass": {
        "c4": 36928,
        "c6": 7096328,
        "discriminant": -19,
        "identity_1728_disc_eq_c4_cubed_minus_c6_squared": true
      },
      "discriminant_factorisation": {
        "19": 1
      },
      "hypotheses": {
        "ordinary at p (a_p not divisible by p)": true,
        "unit root mod p": 3,
        "prime-to-p level (p does not divide N)": true,
        "Hecke polynomial X^2 - a_p X + p has distinct roots (disc nonzero)": true,
        "p-distinguished (the two roots differ mod p)": true,
        "noble (prime-to-p level + distinct Hecke roots), Assumption B1.1": true,
        "non-critical slope (ordinary means slope 0 < r_1 + 1)": true,
        "Hida (ordinary), as section B2 restricts to": true,
        "newform of its level, as section B2 assumes": true,
        "weight 2, trivial nebentypus": true,
        "minimal model (Delta squarefree, and 1728 Delta = c_4^3 - c_6^2)": true,
        "semistable (no additive reduction: every prime dividing Delta fails to divide c_4)": true,
        "residually irreducible at p = 11 (semistable + Mazur: a rational l-isogeny forces l <= 7)": true
      },
      "hecke_polynomial": {
        "discriminant": -35,
        "discriminant_mod_p": 9
      },
      "multiplicative_at_each_bad_prime": {
        "19": true
      },
      "semistable": true,
      "discriminant_squarefree": true,
      "minimal_model_guard": true,
      "all_hold": true
    }
  },
  "smoothing": {
    "d": 5,
    "S": [
      2,
      11,
      19,
      389
    ],
    "6S": 975612,
    "gcd_d_with_6S": 1,
    "holds": true
  },
  "surjectivity_required": false,
  "surjectivity_note": "Mazur's theorem gives irreducibility for free here, and in fact surjectivity, but neither C1.1 nor B1.1 nor section B2 asks for surjectivity; requiring it would be an over-strong hypothesis and the drill plants that as a defect",
  "all_hold": true,
  "agrees": true
}