Elliptic curve data - 49.a4

Formats: - HTML - YAML - JSON - 2025-05-22T04:08:57.138693

• 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': '49a', 'Clabel': '49a1', 'Cnumber': 1, 'abc_quality': 0.9802957926219806, 'absD': 343, 'ainvs': [1, -1, 0, -2, -1], 'analytic_rank': 0, 'bad_primes': [7], 'class_deg': 14, 'class_size': 4, 'cm': -7, 'conductor': 49, 'degree': 1, 'elladic_images': ['7.48.0.6'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.79913838905562241712431170973', 'prec': 107}, 'faltings_index': 0, 'faltings_ratio': 1, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 7, 14], 'jinv': [-3375, 1], 'lmfdb_iso': '49.a', 'lmfdb_label': '49.a4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, -1, 0, -2, -1], 'min_quad_twist_disc': 1, 'modell_images': ['7B.1.5'], 'nonmax_primes': [7], 'nonmax_rad': 7, 'num_bad_primes': 1, 'num_int_pts': 1, 'optimality': 1, 'potential_good_reduction': True, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[7, 20, 1]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-1.2856159263194507434006498956', 'prec': 103}, 'szpiro_ratio': 3.6937251021607387, '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, 1, 0, -1, 0, 0, 0, -3, -3, 0, 4, 0, 0, 0, 0, -1, 0, -3, 0, 0], 'aplist': [1, 0, 0, 0, 4, 0, 0, 0, 8, 2, 0, -6, 0, -12, 0, -10, 0, 0, 4, 16, 0, 8, 0, 0, 0], 'class_deg': 14, 'class_size': 4, 'conductor': 49, 'isogeny_matrix': [[1, 2, 7, 14], [2, 1, 14, 7], [7, 14, 1, 2], [14, 7, 2, 1]], 'lmfdb_iso': '49.a', 'trace_hash': 1094831497817794886}
• 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': '4.9445046002825467364981969682', 'prec': 100}, 'conductor': 49, 'gens': [], 'heights': [], 'lmfdb_label': '49.a4', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '1.9333117056168115467330768390', 'prec': 100}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.96665585280840577336653841951483552682', 'prec': 133}, 'tamagawa_product': 2, 'torsion_generators': [[2, -1, 1]], 'xcoord_integral_points': [2]}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': 'o?', '3': [0, 0, 0], '5': [0, 0, 0], '7': 'a'}, 'iwp0': 11, 'lmfdb_label': '49.a4'}
• ec_torsion_growth •

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

  
  
  • id: 1604
    {'degree': 2, 'field': [2, -1, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 2]}
  • id: 1605
    {'degree': 4, 'field': [-3, -4, 5, -2, 1], 'lmfdb_label': '49.a4', 'torsion': [4]}
  • id: 1606
    {'degree': 4, 'field': [2, -4, 3, -1, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 4]}
  • id: 1607
    {'degree': 6, 'field': [1, -1, 1, -1, 1, -1, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 14]}
  • id: 1610
    {'degree': 8, 'field': [1, -5, 1, -2, 7, -5, 1, -2, 1], 'lmfdb_label': '49.a4', 'torsion': [6]}
  • id: 1608
    {'degree': 8, 'field': [1, -2, 2, 4, -1, 4, 2, -2, 1], 'lmfdb_label': '49.a4', 'torsion': [4, 4]}
  • id: 1609
    {'degree': 8, 'field': [1, 0, -3, 0, 6, 0, -3, 0, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 8]}
  • id: 1611
    {'degree': 12, 'field': [1, -1, -1, -4, 27, -47, 35, -3, -11, 3, 5, -4, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 28]}
  • id: 1615
    {'degree': 16, 'field': [1, -5, 24, -1, -16, 63, -26, -11, 57, -38, 16, 21, -16, 8, 3, -2, 1], 'lmfdb_label': '49.a4', 'torsion': [6, 6]}
  • id: 1612
    {'degree': 16, 'field': [1, -4, -26, 8, 83, -104, -40, 108, -56, -80, -10, 12, 3, -4, -4, 0, 1], 'lmfdb_label': '49.a4', 'torsion': [8]}
  • id: 1613
    {'degree': 16, 'field': [1, 0, 0, 0, -3, 0, 0, 0, 20, 0, 0, 0, -3, 0, 0, 0, 1], 'lmfdb_label': '49.a4', 'torsion': [4, 8]}
  • id: 1614
    {'degree': 16, 'field': [16, 0, -8, 0, 29, 0, 25, 0, 21, 0, 12, 0, 9, 0, 1, 0, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 16]}
  • id: 1616
    {'degree': 20, 'field': [37, -152, 258, -264, 222, -133, 12, 13, 17, 9, -38, 39, -29, -7, 46, -55, 43, -26, 13, -5, 1], 'lmfdb_label': '49.a4', 'torsion': [2, 22]}
  • id: 1617
    {'degree': 21, 'field': [-1, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '49.a4', 'torsion': [14]}
• 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

  
  {'conductor': 49, 'conductor_valuation': 2, 'discriminant_valuation': 3, 'j_denominator_valuation': 0, 'kodaira_symbol': 3, 'lmfdb_label': '49.a4', 'prime': 7, 'reduction_type': 0, '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

  
  {'conductor': 49, 'elladic_image': '7.48.0.6', 'lmfdb_label': '49.a4', 'modell_image': '7B.1.5', 'prime': 7}
• lmfdb_iso 49.a does not appear in ec_padic