Elliptic curve data - 72.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
  intrinsic_torsion	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
  squarefree_disc	boolean
  stable_faltings_height	numeric
  szpiro_ratio	double precision
  torsion	smallint
  torsion_primes	smallint[]
  torsion_structure	smallint[]

  
  {'Ciso': '72a', 'Clabel': '72a2', 'Cnumber': 2, 'abc_quality': 0.972547111469975, 'absD': 1679616, 'adelic_genus': 1, 'adelic_index': 192, 'adelic_level': 24, 'ainvs': [0, 0, 0, -39, -70], 'analytic_rank': 0, 'bad_primes': [2, 3], 'class_deg': 8, 'class_size': 6, 'cm': 0, 'conductor': 72, 'degree': 8, 'elladic_images': ['8.96.0.32'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.096046141745802426961388058364', 'prec': 110}, 'faltings_index': 1, 'faltings_ratio': 2, 'intrinsic_torsion': 1, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4], 'jinv': [35152, 9], 'lmfdb_iso': '72.a', 'lmfdb_label': '72.a4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, -1, 0, -4, 4], 'min_quad_twist_disc': -3, 'modell_images': ['2Cs'], 'modm_images': ['2.6.0.a.1', '4.24.0-4.b.1.2', '8.96.0-8.h.1.7', '12.48.0-12.c.1.4', '24.192.1-24.bu.1.2'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 3, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[2, 2, 9], [3, 8, 8]], 'sha': 1, 'sha_primes': [], 'signD': 1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-1.1074504064531541456038320911', 'prec': 103}, 'szpiro_ratio': 5.285492797907087, 'torsion': 4, 'torsion_primes': [2], 'torsion_structure': [2, 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, 0, 0, 0, 0, -4, 0, -2, 0, 0, 0, -2, 0, -4, 0], 'aplist': [0, 0, 2, 0, -4, -2, -2, -4, 8, -6, 8, 6, 6, 4, 0, 2, -4, -2, -4, -8, 10, -8, 4, 6, 2], 'class_deg': 8, 'class_size': 6, 'conductor': 72, 'isogeny_matrix': [[1, 8, 2, 4, 8, 4], [8, 1, 4, 2, 4, 8], [2, 4, 1, 2, 4, 2], [4, 2, 2, 1, 2, 4], [8, 4, 4, 2, 1, 8], [4, 8, 2, 4, 8, 1]], 'lmfdb_iso': '72.a', 'trace_hash': 1436269620903515173}
• 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': '1.2117823393104796268541644447', 'prec': 100}, 'conductor': 72, 'gens': [], 'heights': [], 'lmfdb_label': '72.a4', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '1.9465368422751189394858646060', 'prec': 100}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.97326842113755946974293230301957677358', 'prec': 133}, 'tamagawa_product': 8, 'torsion_generators': [[-2, 0, 1], [7, 0, 1]], 'xcoord_integral_points': [-5, -2, 7]}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': 'a', '3': 'a'}, 'iwp0': 3, 'lmfdb_label': '72.a4'}
• ec_torsion_growth •

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

  
  
  • id: 2510
    {'degree': 2, 'field': [-3, 0, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 4]}
  • id: 2511
    {'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 4]}
  • id: 2513
    {'degree': 4, 'field': [1, -2, 0, -2, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 8]}
  • id: 2512
    {'degree': 4, 'field': [1, 0, -1, 0, 1], 'lmfdb_label': '72.a4', 'torsion': [4, 4]}
  • id: 2515
    {'degree': 8, 'field': [1, -4, 8, -10, 7, -2, 2, -2, 1], 'lmfdb_label': '72.a4', 'torsion': [4, 8]}
  • id: 2516
    {'degree': 8, 'field': [1, 0, -2, -2, 0, -2, -2, 0, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 6]}
  • id: 2514
    {'degree': 8, 'field': [1, 0, 0, 0, -1, 0, 0, 0, 1], 'lmfdb_label': '72.a4', 'torsion': [4, 8]}
  • id: 2519
    {'degree': 16, 'field': [-2, 16, -40, 0, 158, -220, -34, 280, -138, -116, 124, -10, -30, 14, 2, -4, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 12]}
  • id: 2517
    {'degree': 16, 'field': [1, -12, 68, -232, 488, -564, 194, 396, -665, 444, -62, -152, 162, -92, 34, -8, 1], 'lmfdb_label': '72.a4', 'torsion': [8, 8]}
  • id: 2520
    {'degree': 16, 'field': [1, 0, 2, -4, 4, -2, 0, 8, -9, 8, 0, -2, 4, -4, 2, 0, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 12]}
  • id: 2518
    {'degree': 16, 'field': [4, 0, -32, 0, 64, 0, -80, 0, 76, 0, -56, 0, 28, 0, -8, 0, 1], 'lmfdb_label': '72.a4', 'torsion': [2, 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: 392
    {'conductor': 72, 'conductor_valuation': 3, 'discriminant_valuation': 8, 'j_denominator_valuation': 0, 'kodaira_symbol': -5, 'lmfdb_label': '72.a4', 'prime': 2, 'reduction_type': 0, 'root_number': 1, 'tamagawa_number': 2}
  • id: 393
    {'conductor': 72, 'conductor_valuation': 2, 'discriminant_valuation': 8, 'j_denominator_valuation': 2, 'kodaira_symbol': -6, 'lmfdb_label': '72.a4', 'prime': 3, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 4}
• ec_galrep •

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

  
  
  • id: 1666068
    {'adelic_gens': [[1, 0, 8, 1], [1, 8, 0, 1], [5, 8, 22, 3], [17, 8, 16, 9], [5, 4, 20, 21], [17, 0, 16, 5], [1, 22, 18, 5]], 'adelic_image': '24.192.1.?', 'conductor': 72, 'lmfdb_label': '72.a4', 'prime': 0}
  • id: 226
    {'conductor': 72, 'elladic_image': '8.96.0.32', 'lmfdb_label': '72.a4', 'modell_image': '2Cs', 'prime': 2}
• lmfdb_iso 72.a does not appear in ec_padic