Elliptic curve data - 77.b3

Formats: - HTML - YAML - JSON - 2026-05-13T11:15:08.523002

• ec_curvedata •

  Column	Type
  Ciso	text
  Clabel	text
  Cnumber	smallint
  abc_quality	double precision
  absD	numeric
  adelic_genus	integer
  adelic_index	integer
  adelic_level	bigint
  ainvs	numeric[]
  analytic_rank	smallint
  bad_primes	integer[]
  class_deg	smallint
  class_size	smallint
  cm	smallint
  conductor	integer
  degree	bigint
  elladic_images	text[]
  faltings_height	numeric
  faltings_index	smallint
  faltings_ratio	smallint
  intrinsic_torsion	smallint
  iso_nlabel	smallint
  isogeny_degrees	smallint[]
  jinv	numeric[]
  lmfdb_iso	text
  lmfdb_label	text
  lmfdb_number	smallint
  manin_constant	smallint
  min_quad_twist_ainvs	numeric[]
  min_quad_twist_disc	smallint
  modell_images	text[]
  modm_images	text[]
  nonmax_primes	smallint[]
  nonmax_rad	integer
  num_bad_primes	smallint
  num_int_pts	integer
  optimality	smallint
  potential_good_reduction	boolean
  rank	smallint
  regulator	numeric
  semistable	boolean
  serre_invariants	integer[]
  sha	integer
  sha_primes	smallint[]
  signD	smallint
  squarefree_disc	boolean
  stable_faltings_height	numeric
  szpiro_ratio	double precision
  torsion	smallint
  torsion_primes	smallint[]
  torsion_structure	smallint[]

  
  {'Ciso': '77b', 'Clabel': '77b2', 'Cnumber': 2, 'abc_quality': 1.0659265883154396, 'absD': 115539436859, 'adelic_genus': 3, 'adelic_index': 144, 'adelic_level': 1386, 'ainvs': [0, 1, 1, 441, -15815], 'analytic_rank': 0, 'bad_primes': [7, 11], 'class_deg': 9, 'class_size': 3, 'cm': 0, 'conductor': 77, 'degree': 60, 'elladic_images': ['9.24.0.3'], 'faltings_height': {'__RealLiteral__': 0, 'data': '0.80361261928270555492016418371', 'prec': 103}, 'faltings_index': 2, 'faltings_ratio': 9, 'intrinsic_torsion': 1, 'iso_nlabel': 1, 'isogeny_degrees': [1, 3, 9], 'jinv': [9463555063808, 115539436859], 'lmfdb_iso': '77.b', 'lmfdb_label': '77.b3', 'lmfdb_number': 3, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, 1, 1, 441, -15815], 'min_quad_twist_disc': 1, 'modell_images': ['3B.1.2'], 'modm_images': ['3.8.0-3.a.1.1', '9.24.0-9.a.1.1', '22.2.0.a.1', '63.72.0-63.e.2.2', '66.16.0-66.a.1.1', '198.48.1.?', '1386.144.3.?'], 'nonmax_primes': [3], 'nonmax_rad': 3, 'num_bad_primes': 2, 'num_int_pts': 0, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': True, 'serre_invariants': [[2, 2, 11], [3, 2, 7], [7, 8, 11], [11, 12, 7]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '0.80361261928270555492016418371', 'prec': 103}, 'szpiro_ratio': 7.569178195449453, 'torsion': 1, 'torsion_primes': [], 'torsion_structure': []}
• ec_classdata •

  Column	Type
  anlist	smallint[]
  aplist	smallint[]
  class_deg	smallint
  class_size	smallint
  conductor	integer
  isogeny_matrix	smallint[]
  lmfdb_iso	text
  trace_hash	bigint

  
  {'anlist': [0, 1, 0, 1, -2, 3, 0, 1, 0, -2, 0, -1, -2, -4, 0, 3, 4, -6, 0, 2, -6], 'aplist': [0, 1, 3, 1, -1, -4, -6, 2, 3, -6, 5, 11, 6, 8, 0, -6, -9, -10, 5, 9, 2, -10, 12, -3, -1], 'class_deg': 9, 'class_size': 3, 'conductor': 77, 'isogeny_matrix': [[1, 3, 9], [3, 1, 3], [9, 3, 1]], 'lmfdb_iso': '77.b', 'trace_hash': 874065559975919228}
• ec_mwbsd •

  Column	Type
  area	numeric
  conductor	integer
  gens	numeric[]
  heights	numeric[]
  lmfdb_label	text
  ngens	smallint
  rank_bounds	smallint[]
  real_period	numeric
  sha_an	numeric
  special_value	numeric
  tamagawa_product	integer
  torsion_generators	numeric[]
  xcoord_integral_points	numeric[]

  
  {'area': {'__RealLiteral__': 0, 'data': '0.20044302472760550171235422019', 'prec': 103}, 'conductor': 77, 'gens': [], 'heights': [], 'lmfdb_label': '77.b3', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '0.51630335521161887172962371615', 'prec': 103}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '1.0326067104232377434592474322907733706', 'prec': 130}, 'tamagawa_product': 2, 'torsion_generators': [], 'xcoord_integral_points': []}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': [2, 3, 0], '3': [0, 2], '7': [1, 0], '11': [0, 0]}, 'iwp0': 5, 'lmfdb_label': '77.b3'}
• ec_torsion_growth •

  Column	Type
  degree	smallint
  field	numeric[]
  lmfdb_label	text
  torsion	smallint[]

  
  
  • id: 2771
    {'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '77.b3', 'torsion': [3]}
  • id: 2773
    {'degree': 3, 'field': [-7, 0, 0, 1], 'lmfdb_label': '77.b3', 'torsion': [3]}
  • id: 2772
    {'degree': 3, 'field': [1, 1, -1, 1], 'lmfdb_label': '77.b3', 'torsion': [2]}
  • id: 2774
    {'degree': 6, 'field': [1, -1, 2, -3, 2, -1, 1], 'lmfdb_label': '77.b3', 'torsion': [2, 2]}
  • id: 2778
    {'degree': 6, 'field': [1, 1, 2, -3, 0, -1, 1], 'lmfdb_label': '77.b3', 'torsion': [6]}
  • id: 2777
    {'degree': 6, 'field': [7, 0, 0, -5, 0, 0, 1], 'lmfdb_label': '77.b3', 'torsion': [9]}
  • id: 2775
    {'degree': 6, 'field': [64, -24, -15, 7, 6, -3, 1], 'lmfdb_label': '77.b3', 'torsion': [3, 3]}
  • id: 2776
    {'degree': 6, 'field': [343, 0, 0, -35, 0, 0, 1], 'lmfdb_label': '77.b3', 'torsion': [9]}
  • id: 2779
    {'degree': 9, 'field': [-343, 0, 0, 343, 0, 0, -35, 0, 0, 1], 'lmfdb_label': '77.b3', 'torsion': [6]}
  • id: 2780
    {'degree': 12, 'field': [-83, -70, 40, -192, 209, -64, -8, 24, -5, -10, 8, -4, 1], 'lmfdb_label': '77.b3', 'torsion': [4]}
  • id: 2781
    {'degree': 12, 'field': [1, -1, -1, 4, -1, -3, 6, -3, -1, 4, -1, -1, 1], 'lmfdb_label': '77.b3', 'torsion': [2, 6]}
  • id: 2785
    {'degree': 18, 'field': [343, 0, 0, -1715, 0, 0, 2786, 0, 0, -1245, 0, 0, 252, 0, 0, -25, 0, 0, 1], 'lmfdb_label': '77.b3', 'torsion': [18]}
  • id: 2782
    {'degree': 18, 'field': [811, -5190, 14379, -20937, 16131, -4305, -4059, 7515, -5535, 2354, 264, -1407, 1359, -861, 411, -156, 45, -9, 1], 'lmfdb_label': '77.b3', 'torsion': [3, 9]}
  • id: 2783
    {'degree': 18, 'field': [21316, 8760, -45456, 115644, 130326, -117804, 111709, 23943, -23067, 14176, 1503, -2187, 982, 87, -81, 32, 3, -3, 1], 'lmfdb_label': '77.b3', 'torsion': [3, 6]}
  • id: 2786
    {'degree': 18, 'field': [117649, 0, 0, -67228, 0, 0, 26411, 0, 0, -4116, 0, 0, 539, 0, 0, -28, 0, 0, 1], 'lmfdb_label': '77.b3', 'torsion': [2, 6]}
  • id: 2784
    {'degree': 18, 'field': [65624329, -19103799, -7859073, 10976696, -248745, -2332161, 1271259, 98013, -196284, 49770, 22887, -13587, 1775, 1329, -465, 61, 21, -6, 1], 'lmfdb_label': '77.b3', 'torsion': [18]}
• ec_localdata •

  Column	Type
  conductor	integer
  conductor_valuation	smallint
  discriminant_valuation	smallint
  j_denominator_valuation	smallint
  kodaira_symbol	smallint
  lmfdb_label	text
  prime	integer
  reduction_type	smallint
  root_number	smallint
  tamagawa_number	smallint

  
  
  • id: 434
    {'conductor': 77, 'conductor_valuation': 1, 'discriminant_valuation': 2, 'j_denominator_valuation': 2, 'kodaira_symbol': 6, 'lmfdb_label': '77.b3', 'prime': 7, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 2}
  • id: 435
    {'conductor': 77, 'conductor_valuation': 1, 'discriminant_valuation': 9, 'j_denominator_valuation': 9, 'kodaira_symbol': 13, 'lmfdb_label': '77.b3', 'prime': 11, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 1}
• ec_galrep •

  Column	Type
  adelic_gens	bigint[]
  adelic_image	text
  conductor	integer
  elladic_image	text
  lmfdb_label	text
  modell_image	text
  prime	smallint

  
  
  • id: 1666089
    {'adelic_gens': [[1, 18, 0, 1], [10, 9, 81, 73], [1, 0, 18, 1], [1369, 18, 1368, 19], [1, 18, 0, 155], [262, 9, 873, 1378], [1, 18, 10, 181], [7, 18, 1170, 463]], 'adelic_image': '1386.144.3.?', 'conductor': 77, 'lmfdb_label': '77.b3', 'prime': 0}
  • id: 245
    {'conductor': 77, 'elladic_image': '9.24.0.3', 'lmfdb_label': '77.b3', 'modell_image': '3B.1.2', 'prime': 3}
• lmfdb_iso 77.b does not appear in ec_padic