Formats: - HTML - YAML - JSON - 2026-07-21T01:25:16.529295
  • ec_curvedata • Show schema
    {'Ciso': '27a', 'Clabel': '27a2', 'Cnumber': 2, 'abc_quality': 1.23864473399791, 'absD': 177147, 'ainvs': [0, 0, 1, -270, -1708], 'analytic_rank': 0, 'bad_primes': [3], 'class_deg': 27, 'class_size': 4, 'cm': -27, 'conductor': 27, 'degree': 3, 'elladic_images': ['27.648.13.34'], 'faltings_height': {'__RealLiteral__': 0, 'data': '0.052147932407099199254187350298', 'prec': 107}, 'faltings_index': 3, 'faltings_ratio': 9, 'intrinsic_torsion': 1, 'iso_nlabel': 0, 'isogeny_degrees': [1, 3, 9, 27], 'jinv': [-12288000, 1], 'lmfdb_iso': '27.a', 'lmfdb_label': '27.a1', 'lmfdb_number': 1, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, 0, 1, -30, 63], 'min_quad_twist_disc': -3, 'modell_images': ['3B.1.2'], 'nonmax_primes': [3], 'nonmax_rad': 3, 'num_bad_primes': 1, 'num_int_pts': 0, 'optimality': 0, 'potential_good_reduction': True, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[3, 4, 1]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-0.95491333220533468452478745021', 'prec': 107}, 'szpiro_ratio': 8.619664953039187, 'torsion': 1, 'torsion_primes': [], 'torsion_structure': []}
  • ec_classdata • Show schema
    {'anlist': [0, 1, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, 4, 0, 0, -7, 0], 'aplist': [0, 0, 0, -1, 0, 5, 0, -7, 0, 0, -4, 11, 0, 8, 0, 0, 0, -1, 5, 0, -7, 17, 0, 0, -19], 'class_deg': 27, 'class_size': 4, 'conductor': 27, 'isogeny_matrix': [[1, 27, 3, 9], [27, 1, 9, 3], [3, 9, 1, 3], [9, 3, 3, 1]], 'lmfdb_iso': '27.a', 'trace_hash': 903291523394155081}
  • ec_mwbsd • Show schema
    {'area': {'__RealLiteral__': 0, 'data': '0.90095869600696607120514478004', 'prec': 103}, 'conductor': 27, 'gens': [], 'heights': [], 'lmfdb_label': '27.a1', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '0.58887958342848331910456316655', 'prec': 103}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.58887958342848331910456316654932546833', 'prec': 133}, 'tamagawa_product': 1, 'torsion_generators': [], 'xcoord_integral_points': []}
  • ec_iwasawa • Show schema
    {'iwdata': {'2': [0, 5, 0], '3': 'a'}, 'iwp0': 5, 'lmfdb_label': '27.a1'}
  • ec_torsion_growth • Show schema
    • id: 639
      {'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '27.a1', 'torsion': [3]}
    • id: 641
      {'degree': 3, 'field': [-3, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [3]}
    • id: 640
      {'degree': 3, 'field': [-2, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [2]}
    • id: 642
      {'degree': 6, 'field': [1, -3, 0, 5, 0, -3, 1], 'lmfdb_label': '27.a1', 'torsion': [2, 6]}
    • id: 644
      {'degree': 6, 'field': [1, 0, 0, -1, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [9]}
    • id: 645
      {'degree': 6, 'field': [3, 0, 0, -3, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [9]}
    • id: 643
      {'degree': 6, 'field': [3, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [3, 3]}
    • id: 646
      {'degree': 9, 'field': [1, 0, 0, 3, 0, 0, -3, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [6]}
    • id: 647
      {'degree': 12, 'field': [-2, 0, 12, 4, -18, -12, 0, 12, 12, -2, -6, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [4]}
    • id: 648
      {'degree': 12, 'field': [1, -6, 21, -44, 66, -81, 84, -72, 54, -35, 18, -6, 1], 'lmfdb_label': '27.a1', 'torsion': [21]}
    • id: 651
      {'degree': 18, 'field': [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [27]}
    • id: 650
      {'degree': 18, 'field': [1, 0, 0, 0, 0, 0, 27, 0, 0, -53, 0, 0, 36, 0, 0, -9, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [27]}
    • id: 654
      {'degree': 18, 'field': [1, 0, 0, 3, 0, 0, 12, 0, 0, -11, 0, 0, 6, 0, 0, -3, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [2, 18]}
    • id: 649
      {'degree': 18, 'field': [1, 0, 0, 6, 0, 0, 33, 0, 0, 20, 0, 0, 15, 0, 0, -3, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [3, 9]}
    • id: 653
      {'degree': 18, 'field': [3, 0, -9, 0, 18, -9, -6, 0, 0, 27, -63, 72, -21, -63, 105, -81, 36, -9, 1], 'lmfdb_label': '27.a1', 'torsion': [2, 18]}
    • id: 652
      {'degree': 18, 'field': [27, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 45, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '27.a1', 'torsion': [6, 6]}
  • ec_localdata • Show schema
    {'conductor': 27, 'conductor_valuation': 3, 'discriminant_valuation': 11, 'j_denominator_valuation': 0, 'kodaira_symbol': -2, 'lmfdb_label': '27.a1', 'prime': 3, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 1}
  • ec_galrep • Show schema
    {'conductor': 27, 'elladic_image': '27.648.13.34', 'lmfdb_label': '27.a1', 'modell_image': '3B.1.2', 'prime': 3}
  • lmfdb_iso 27.a does not appear in ec_padic