Elliptic curve data - 50.a1

Formats: - HTML - YAML - JSON - 2025-09-12T08:50:21.555193

• 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': '50a', 'Clabel': '50a2', 'Cnumber': 2, 'abc_quality': 1.050775872125697, 'absD': 5000, 'adelic_genus': 9, 'adelic_index': 384, 'adelic_level': 120, 'ainvs': [1, 0, 1, -126, -552], 'analytic_rank': 0, 'bad_primes': [2, 5], 'class_deg': 15, 'class_size': 4, 'cm': 0, 'conductor': 50, 'degree': 6, 'elladic_images': ['8.2.0.1', '3.8.0.2', '5.24.0.4'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.17679345180982668705620276153', 'prec': 107}, 'faltings_index': 1, 'faltings_ratio': 3, 'iso_nlabel': 0, 'isogeny_degrees': [1, 3, 5, 15], 'jinv': [-349938025, 8], 'lmfdb_iso': '50.a', 'lmfdb_label': '50.a1', 'lmfdb_number': 1, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, 0, 1, -126, -552], 'min_quad_twist_disc': 1, 'modell_images': ['3B.1.2', '5B.1.3'], 'modm_images': ['3.8.0-3.a.1.1', '5.24.0-5.a.2.1', '8.2.0.a.1', '15.192.1-15.a.1.1', '24.16.0-24.a.1.6', '40.48.1-40.bx.2.5', '120.384.9.?'], 'nonmax_primes': [2, 3, 5], 'nonmax_rad': 30, 'num_bad_primes': 2, 'num_int_pts': 0, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[2, 4, 25], [3, 2, 25], [5, 14, 2]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-0.71327275595452681192312253927', 'prec': 107}, 'szpiro_ratio': 6.674566520238682, '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, -1, 1, 1, 0, -1, 2, -1, -2, 0, -3, 1, -4, -2, 0, 1, -3, 2, 5, 0], 'aplist': [-1, 1, 0, 2, -3, -4, -3, 5, 6, 0, 2, 2, -3, -4, 12, 6, 0, 2, -13, 12, 11, -10, -9, 15, 2], 'class_deg': 15, 'class_size': 4, 'conductor': 50, 'isogeny_matrix': [[1, 15, 3, 5], [15, 1, 5, 3], [3, 5, 1, 15], [5, 3, 15, 1]], 'lmfdb_iso': '50.a', 'trace_hash': 1291586430715429255}
• 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': '1.4241667468189884754547775645', 'prec': 100}, 'conductor': 50, 'gens': [], 'heights': [], 'lmfdb_label': '50.a1', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '0.71316498142829675600526799003', 'prec': 103}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.71316498142829675600526799002803169498', 'prec': 133}, 'tamagawa_product': 1, 'torsion_generators': [], 'xcoord_integral_points': []}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': [2, 0], '3': [2, 1], '5': 'a'}, 'iwp0': 7, 'lmfdb_label': '50.a1'}
• ec_torsion_growth •

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

  
  
  • id: 1678
    {'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '50.a1', 'torsion': [3]}
  • id: 1680
    {'degree': 3, 'field': [-5, 0, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [3]}
  • id: 1679
    {'degree': 3, 'field': [2, 2, -1, 1], 'lmfdb_label': '50.a1', 'torsion': [2]}
  • id: 1681
    {'degree': 4, 'field': [1, -1, 1, -1, 1], 'lmfdb_label': '50.a1', 'torsion': [5]}
  • id: 1682
    {'degree': 6, 'field': [2, 0, -3, 0, 4, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [2, 2]}
  • id: 1684
    {'degree': 6, 'field': [4, 4, 6, -6, -1, -1, 1], 'lmfdb_label': '50.a1', 'torsion': [6]}
  • id: 1683
    {'degree': 6, 'field': [36, -18, -9, 3, 6, -3, 1], 'lmfdb_label': '50.a1', 'torsion': [3, 3]}
  • id: 1685
    {'degree': 8, 'field': [1, -1, 0, 1, -1, 1, 0, -1, 1], 'lmfdb_label': '50.a1', 'torsion': [15]}
  • id: 1686
    {'degree': 9, 'field': [1, 0, 0, 13, 0, 0, -7, 0, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [6]}
  • id: 1687
    {'degree': 10, 'field': [-1, 5, -5, -10, 35, -49, 45, -30, 15, -5, 1], 'lmfdb_label': '50.a1', 'torsion': [5]}
  • id: 1689
    {'degree': 12, 'field': [1, -1, 4, -10, 5, 14, -19, 1, 15, -15, 9, -4, 1], 'lmfdb_label': '50.a1', 'torsion': [10]}
  • id: 1688
    {'degree': 12, 'field': [2, 2, -15, 0, 27, 0, 4, 8, 6, -2, -1, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [4]}
  • id: 1690
    {'degree': 12, 'field': [4, 0, 6, 0, 1, 0, 16, 0, 19, 0, -4, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [2, 6]}
  • id: 1691
    {'degree': 12, 'field': [5, 0, 0, -15, 0, 0, 15, 0, 0, -5, 0, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [15]}
  • id: 1692
    {'degree': 18, 'field': [1, 0, 0, -8, 0, 0, 11, 0, 0, 28, 0, 0, 11, 0, 0, -8, 0, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [9]}
  • id: 1694
    {'degree': 18, 'field': [8, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '50.a1', 'torsion': [2, 6]}
  • id: 1693
    {'degree': 18, 'field': [400, -1800, 4500, -6720, 6450, -4440, 3109, -3177, 3465, -2924, 1821, -969, 628, -483, 309, -142, 45, -9, 1], 'lmfdb_label': '50.a1', 'torsion': [3, 6]}
  • id: 1695
    {'degree': 20, 'field': [1, -5, 5, 15, -30, -21, 75, 15, -120, -5, 141, -5, -120, 15, 75, -21, -30, 15, 5, -5, 1], 'lmfdb_label': '50.a1', 'torsion': [5, 5]}
  • id: 1696
    {'degree': 20, 'field': [1, -5, 10, -15, 30, -53, 50, -15, 45, -200, 308, -200, 45, -15, 50, -53, 30, -15, 10, -5, 1], 'lmfdb_label': '50.a1', 'torsion': [15]}
• 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: 233
    {'conductor': 50, 'conductor_valuation': 1, 'discriminant_valuation': 3, 'j_denominator_valuation': 3, 'kodaira_symbol': 7, 'lmfdb_label': '50.a1', 'prime': 2, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 1}
  • id: 234
    {'conductor': 50, 'conductor_valuation': 2, 'discriminant_valuation': 4, 'j_denominator_valuation': 0, 'kodaira_symbol': 4, 'lmfdb_label': '50.a1', 'prime': 5, 'reduction_type': 0, '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: 1665994
    {'adelic_gens': [[73, 90, 0, 49], [1, 0, 60, 1], [41, 80, 40, 81], [1, 36, 60, 1], [31, 90, 15, 91], [46, 105, 45, 1], [41, 90, 40, 1], [1, 0, 30, 1], [61, 90, 45, 91], [1, 72, 0, 1], [1, 42, 90, 61]], 'adelic_image': '120.384.9.?', 'conductor': 50, 'lmfdb_label': '50.a1', 'prime': 0}
  • id: 132
    {'conductor': 50, 'elladic_image': '8.2.0.1', 'lmfdb_label': '50.a1', 'modell_image': '2G', 'prime': 2}
  • id: 133
    {'conductor': 50, 'elladic_image': '3.8.0.2', 'lmfdb_label': '50.a1', 'modell_image': '3B.1.2', 'prime': 3}
  • id: 134
    {'conductor': 50, 'elladic_image': '5.24.0.4', 'lmfdb_label': '50.a1', 'modell_image': '5B.1.3', 'prime': 5}
• lmfdb_iso 50.a does not appear in ec_padic