Elliptic curve data - 56.a2

Formats: - HTML - YAML - JSON - 2024-07-27T12:53:31.169294

• 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
  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
  sha	integer
  sha_primes	smallint[]
  signD	smallint
  stable_faltings_height	numeric
  szpiro_ratio	double precision
  torsion	smallint
  torsion_primes	smallint[]
  torsion_structure	smallint[]

  
  {'Ciso': '56a', 'Clabel': '56a3', 'Cnumber': 3, 'abc_quality': 1.1170620055320515, 'absD': 4917248, 'adelic_genus': 0, 'adelic_index': 48, 'adelic_level': 56, 'ainvs': [0, 0, 0, -59, -138], 'analytic_rank': 0, 'bad_primes': [2, 7], 'class_deg': 4, 'class_size': 4, 'cm': 0, 'conductor': 56, 'degree': 8, 'elladic_images': ['8.24.0.58'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.0020328267985874883465435369944', 'prec': 113}, 'faltings_index': 2, 'faltings_ratio': 4, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4], 'jinv': [11090466, 2401], 'lmfdb_iso': '56.a', 'lmfdb_label': '56.a2', 'lmfdb_number': 2, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, 0, 0, -59, -138], 'min_quad_twist_disc': 1, 'modell_images': ['2B'], 'modm_images': ['2.3.0.a.1', '4.12.0-4.c.1.2', '8.24.0-8.k.1.1', '56.48.0-56.v.1.8'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 1, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': False, 'sha': 1, 'sha_primes': [], 'signD': 1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-0.63741774231187068864567298166', 'prec': 107}, 'szpiro_ratio': 5.924007936828446, 'torsion': 2, 'torsion_primes': [2], 'torsion_structure': [2]}
• 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, 0, 0, 2, 0, -1, 0, -3, 0, -4, 0, 2, 0, 0, 0, -6, 0, 8, 0], 'aplist': [0, 0, 2, -1, -4, 2, -6, 8, 0, 6, 8, -2, 2, -4, -8, 6, 0, -6, -4, -8, 10, 16, 8, -6, -6], 'class_deg': 4, 'class_size': 4, 'conductor': 56, 'isogeny_matrix': [[1, 4, 2, 4], [4, 1, 2, 4], [2, 2, 1, 2], [4, 4, 2, 1]], 'lmfdb_iso': '56.a', 'trace_hash': 1158293942075606764}
• 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': '1.0040739295787181528074062214', 'prec': 100}, 'conductor': 56, 'gens': [], 'heights': [], 'lmfdb_label': '56.a2', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '1.7490966283766727306017206995', 'prec': 100}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.87454831418833636530086034972735623668', 'prec': 133}, 'tamagawa_product': 2, 'torsion_generators': [[-6, 0, 1]], 'xcoord_integral_points': [-6]}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': 'a', '7': [0, 0]}, 'iwp0': 3, 'lmfdb_label': '56.a2'}
• ec_torsion_growth •

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

  
  
  • id: 1960
    {'degree': 2, 'field': [-2, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 2]}
  • id: 1961
    {'degree': 2, 'field': [1, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [4]}
  • id: 1962
    {'degree': 2, 'field': [2, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [4]}
  • id: 1964
    {'degree': 4, 'field': [-2, 0, 0, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 4]}
  • id: 1963
    {'degree': 4, 'field': [1, 0, 0, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 4]}
  • id: 1968
    {'degree': 8, 'field': [1, -4, 4, 2, -8, 2, 4, -4, 1], 'lmfdb_label': '56.a2', 'torsion': [6]}
  • id: 1965
    {'degree': 8, 'field': [1, 0, -4, 0, 8, 0, -4, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [4, 4]}
  • id: 1966
    {'degree': 8, 'field': [1, 0, 1, 0, 4, 0, 1, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [8]}
  • id: 1967
    {'degree': 8, 'field': [16, 0, 0, 0, -6, 0, 0, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [8]}
  • id: 1969
    {'degree': 16, 'field': [1, 0, 0, 0, -7, 0, 0, 0, 16, 0, 0, 0, -7, 0, 0, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 8]}
  • id: 1971
    {'degree': 16, 'field': [4, 0, 0, 0, -32, 0, 0, 0, 36, 0, 0, 0, -16, 0, 0, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 8]}
  • id: 1973
    {'degree': 16, 'field': [9, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 12, 0, 0, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [12]}
  • id: 1970
    {'degree': 16, 'field': [16, 0, -32, 0, 32, 0, -88, 0, 161, 0, -44, 0, 8, 0, -4, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 8]}
  • id: 1972
    {'degree': 16, 'field': [64, 0, -256, 0, 256, 0, 128, 0, -176, 0, -40, 0, 4, 0, -4, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [2, 6]}
  • id: 1974
    {'degree': 16, 'field': [64, 0, 256, 0, 256, 0, -128, 0, -176, 0, 40, 0, 4, 0, 4, 0, 1], 'lmfdb_label': '56.a2', 'torsion': [12]}
• 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: 280
    {'conductor': 56, 'conductor_valuation': 3, 'discriminant_valuation': 11, 'j_denominator_valuation': 0, 'kodaira_symbol': -2, 'lmfdb_label': '56.a2', 'prime': 2, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 1}
  • id: 281
    {'conductor': 56, 'conductor_valuation': 1, 'discriminant_valuation': 4, 'j_denominator_valuation': 4, 'kodaira_symbol': 8, 'lmfdb_label': '56.a2', 'prime': 7, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}
• ec_galrep •

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

  
  
  • id: 1666018
    {'adelic_gens': [[17, 8, 12, 33], [49, 8, 48, 9], [1, 0, 8, 1], [1, 4, 4, 17], [7, 6, 50, 51], [43, 36, 26, 13], [24, 1, 27, 14], [1, 8, 0, 1]], 'adelic_image': '56.48.0.?', 'conductor': 56, 'lmfdb_label': '56.a2', 'prime': 0}
  • id: 171
    {'conductor': 56, 'elladic_image': '8.24.0.58', 'lmfdb_label': '56.a2', 'modell_image': '2B', 'prime': 2}
• lmfdb_iso 56.a does not appear in ec_padic