Elliptic curve data - 21.a4

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• 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': '21a', 'Clabel': '21a6', 'Cnumber': 6, 'abc_quality': 1.042661241253942, 'absD': 17294403, 'adelic_genus': 1, 'adelic_index': 192, 'adelic_level': 336, 'ainvs': [1, 0, 0, -34, -217], 'analytic_rank': 0, 'bad_primes': [3, 7], 'class_deg': 8, 'class_size': 6, 'cm': 0, 'conductor': 21, 'degree': 4, 'elladic_images': ['8.48.0.177'], 'faltings_height': {'__RealLiteral__': 0, 'data': '0.074205383390939770758265854672', 'prec': 107}, 'faltings_index': 4, 'faltings_ratio': 8, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4, 8], 'jinv': [-4354703137, 17294403], 'lmfdb_iso': '21.a', 'lmfdb_label': '21.a4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, 0, 0, -34, -217], 'min_quad_twist_disc': 1, 'modell_images': ['2B'], 'modm_images': ['2.3.0.a.1', '4.12.0-4.c.1.2', '6.6.0.a.1', '8.48.0-8.bb.2.3', '12.24.0-12.g.1.1', '24.96.0-24.bj.1.1', '112.96.0.?', '336.192.1.?'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 0, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': True, 'serre_invariants': [[2, 2, 3], [3, 4, 7], [7, 8, 3]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '0.074205383390939770758265854672', 'prec': 107}, 'szpiro_ratio': 7.967306624104166, '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, 1, -1, -2, -1, -1, 3, 1, 2, 4, -1, -2, 1, -2, -1, -6, -1, 4, 2], 'aplist': [-1, 1, -2, -1, 4, -2, -6, 4, 0, -2, 0, 6, 2, -4, 0, 6, 12, -2, 4, 0, -6, -16, -12, -14, 18], 'class_deg': 8, 'class_size': 6, 'conductor': 21, 'isogeny_matrix': [[1, 2, 8, 4, 4, 8], [2, 1, 4, 2, 2, 4], [8, 4, 1, 8, 2, 4], [4, 2, 8, 1, 4, 8], [4, 2, 2, 4, 1, 2], [8, 4, 4, 8, 2, 1]], 'lmfdb_iso': '21.a', 'trace_hash': 1161919666425559947}
• 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': '0.86207692963712698063615726709', 'prec': 103}, 'conductor': 21, 'gens': [], 'heights': [], 'lmfdb_label': '21.a4', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '0.90223081077698411118680031281', 'prec': 103}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.45111540538849205559340015640263020881', 'prec': 133}, 'tamagawa_product': 2, 'torsion_generators': [[31, -31, 2]], 'xcoord_integral_points': []}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': [1, 2], '3': [1, 0], '7': [0, 0]}, 'iwp0': 3, 'lmfdb_label': '21.a4'}
• ec_torsion_growth •

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

  
  
  • id: 472
    {'degree': 2, 'field': [-3, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [4]}
  • id: 471
    {'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 2]}
  • id: 473
    {'degree': 2, 'field': [1, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [4]}
  • id: 476
    {'degree': 4, 'field': [-3, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [8]}
  • id: 474
    {'degree': 4, 'field': [1, 0, -1, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 4]}
  • id: 478
    {'degree': 4, 'field': [1, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [8]}
  • id: 475
    {'degree': 4, 'field': [3, 0, -3, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 4]}
  • id: 477
    {'degree': 4, 'field': [9, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [8]}
  • id: 483
    {'degree': 8, 'field': [-3, 0, -1, 0, -6, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [6]}
  • id: 479
    {'degree': 8, 'field': [1, -4, 8, -10, 7, -2, 2, -2, 1], 'lmfdb_label': '21.a4', 'torsion': [4, 4]}
  • id: 480
    {'degree': 8, 'field': [1, 0, 0, 0, -1, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 8]}
  • id: 482
    {'degree': 8, 'field': [3, 0, 0, 0, -3, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 8]}
  • id: 481
    {'degree': 8, 'field': [9, 0, 0, 0, 3, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 8]}
  • id: 484
    {'degree': 16, 'field': [1, -12, 68, -232, 488, -564, 194, 396, -665, 444, -62, -152, 162, -92, 34, -8, 1], 'lmfdb_label': '21.a4', 'torsion': [4, 8]}
  • id: 490
    {'degree': 16, 'field': [9, 0, -9, 0, -99, 0, -285, 0, 177, 0, -24, 0, -14, 0, 5, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [12]}
  • id: 489
    {'degree': 16, 'field': [9, 0, -3, 0, -17, 0, -6, 0, 39, 0, -2, 0, 6, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 6]}
  • id: 485
    {'degree': 16, 'field': [9, 0, 0, 0, 54, 0, 0, 0, 87, 0, 0, 0, 6, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [4, 8]}
  • id: 486
    {'degree': 16, 'field': [9, 0, 0, 0, 360, 0, 0, 0, 507, 0, 0, 0, -120, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [16]}
  • id: 491
    {'degree': 16, 'field': [9, 0, 72, 0, 144, 0, 31, 0, 80, 0, -23, 0, 1, 0, 2, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [12]}
  • id: 488
    {'degree': 16, 'field': [1296, 0, 0, 0, -864, 0, 0, 0, 265, 0, 0, 0, -24, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [16]}
  • id: 487
    {'degree': 16, 'field': [104976, 0, 0, 0, -23328, 0, 0, 0, 2385, 0, 0, 0, -72, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [16]}
• 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: 56
    {'conductor': 21, 'conductor_valuation': 1, 'discriminant_valuation': 1, 'j_denominator_valuation': 1, 'kodaira_symbol': 5, 'lmfdb_label': '21.a4', 'prime': 3, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 1}
  • id: 57
    {'conductor': 21, 'conductor_valuation': 1, 'discriminant_valuation': 8, 'j_denominator_valuation': 8, 'kodaira_symbol': 12, 'lmfdb_label': '21.a4', '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: 1665919
    {'adelic_gens': [[5, 4, 332, 333], [241, 16, 248, 129], [1, 16, 0, 1], [232, 1, 191, 10], [15, 2, 238, 323], [55, 16, 54, 73], [321, 16, 320, 17], [1, 0, 16, 1], [13, 16, 164, 105]], 'adelic_image': '336.192.1.?', 'conductor': 21, 'lmfdb_label': '21.a4', 'prime': 0}
  • id: 41
    {'conductor': 21, 'elladic_image': '8.48.0.177', 'lmfdb_label': '21.a4', 'modell_image': '2B', 'prime': 2}
• lmfdb_iso 21.a does not appear in ec_padic