Elliptic curve data - 56.a4

Formats: - HTML - YAML - JSON - 2025-07-15T17:01:54.044431

• 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
  serre_invariants	integer[]
  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': '56a1', 'Cnumber': 1, 'abc_quality': 0.8915192755066496, 'absD': 1792, 'adelic_genus': 0, 'adelic_index': 48, 'adelic_level': 56, 'ainvs': [0, 0, 0, 1, 2], 'analytic_rank': 0, 'bad_primes': [2, 7], 'class_deg': 4, 'class_size': 4, 'cm': 0, 'conductor': 56, 'degree': 2, 'elladic_images': ['8.24.0.50'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.69518000735853279776377565845', 'prec': 107}, 'faltings_index': 0, 'faltings_ratio': 1, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4], 'jinv': [432, 7], 'lmfdb_iso': '56.a', 'lmfdb_label': '56.a4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, 0, 0, 1, 2], 'min_quad_twist_disc': 1, 'modell_images': ['2B'], 'modm_images': ['2.3.0.a.1', '4.12.0-4.c.1.1', '8.24.0-8.p.1.1', '14.6.0.b.1', '28.24.0-28.g.1.2', '56.48.0-56.bn.1.6'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 3, 'optimality': 1, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[2, 2, 7], [7, 8, 8]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-1.1572781277318296707085970728', 'prec': 103}, 'szpiro_ratio': 3.7038850376835257, 'torsion': 4, 'torsion_primes': [2], 'torsion_structure': [4]}
• 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': '4.0162957183148726112296248857', 'prec': 100}, 'conductor': 56, 'gens': [], 'heights': [], 'lmfdb_label': '56.a4', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '3.4981932567533454612034413989', 'prec': 100}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.87454831418833636530086034972735623668', 'prec': 133}, 'tamagawa_product': 4, 'torsion_generators': [[1, 2, 1]], 'xcoord_integral_points': [-1, 1]}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': 'a', '7': [0, 0]}, 'iwp0': 3, 'lmfdb_label': '56.a4'}
• ec_torsion_growth •

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

  
  
  • id: 1943
    {'degree': 2, 'field': [2, -1, 1], 'lmfdb_label': '56.a4', 'torsion': [2, 4]}
  • id: 1944
    {'degree': 4, 'field': [-2, -4, -2, 0, 1], 'lmfdb_label': '56.a4', 'torsion': [8]}
  • id: 1948
    {'degree': 8, 'field': [1, -4, 4, 2, -8, 2, 4, -4, 1], 'lmfdb_label': '56.a4', 'torsion': [12]}
  • id: 1945
    {'degree': 8, 'field': [1, -2, 2, 4, -1, 4, 2, -2, 1], 'lmfdb_label': '56.a4', 'torsion': [4, 4]}
  • id: 1946
    {'degree': 8, 'field': [1, 0, -10, 0, 34, 0, -10, 0, 1], 'lmfdb_label': '56.a4', 'torsion': [2, 8]}
  • id: 1947
    {'degree': 8, 'field': [1, 0, 2, 0, 10, 0, 2, 0, 1], 'lmfdb_label': '56.a4', 'torsion': [2, 8]}
  • id: 1949
    {'degree': 16, 'field': [4, 32, 96, 144, 152, 192, 216, 128, 36, 16, -16, -56, -44, -16, -4, 0, 1], 'lmfdb_label': '56.a4', 'torsion': [16]}
  • id: 1950
    {'degree': 16, 'field': [9, 0, -36, 0, 324, 0, -72, 0, 30, 0, 12, 0, -12, 0, 0, 0, 1], 'lmfdb_label': '56.a4', 'torsion': [2, 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: 276
    {'conductor': 56, 'conductor_valuation': 3, 'discriminant_valuation': 8, 'j_denominator_valuation': 0, 'kodaira_symbol': -5, 'lmfdb_label': '56.a4', 'prime': 2, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 4}
  • id: 277
    {'conductor': 56, 'conductor_valuation': 1, 'discriminant_valuation': 1, 'j_denominator_valuation': 1, 'kodaira_symbol': 5, 'lmfdb_label': '56.a4', 'prime': 7, '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: 1666020
    {'adelic_gens': [[43, 38, 26, 51], [25, 24, 6, 31], [49, 8, 48, 9], [24, 3, 29, 2], [1, 0, 8, 1], [1, 4, 4, 17], [7, 6, 50, 51], [1, 8, 0, 1]], 'adelic_image': '56.48.0.?', 'conductor': 56, 'lmfdb_label': '56.a4', 'prime': 0}
  • id: 173
    {'conductor': 56, 'elladic_image': '8.24.0.50', 'lmfdb_label': '56.a4', 'modell_image': '2B', 'prime': 2}
• lmfdb_iso 56.a does not appear in ec_padic