Elliptic curve data - 1728.x1

Formats: - HTML - YAML - JSON - 2026-10-15T10:43:14.444208

• 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': '1728q', 'Clabel': '1728q1', 'Cnumber': 1, 'abc_quality': 1.2262943855309167, 'absD': 1259712, 'adelic_genus': 1, 'adelic_index': 24, 'adelic_level': 24, 'ainvs': [0, 0, 0, -54, 162], 'analytic_rank': 0, 'bad_primes': [2, 3], 'class_deg': 1, 'class_size': 1, 'cm': 0, 'conductor': 1728, 'degree': 288, 'elladic_images': ['3.3.0.1'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.079872223891064687667185569069', 'prec': 97}, 'faltings_index': 0, 'faltings_ratio': 1, 'intrinsic_torsion': 1, 'iso_nlabel': 23, 'isogeny_degrees': [1], 'jinv': [-13824, 1], 'lmfdb_iso': '1728.x', 'lmfdb_label': '1728.x1', 'lmfdb_number': 1, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, 0, 0, -216, -1296], 'min_quad_twist_disc': -8, 'modell_images': ['3Nn'], 'modm_images': ['3.3.0.a.1', '6.6.1.b.1', '12.12.1.g.1', '24.24.1-12.g.1.1'], 'nonmax_primes': [3], 'nonmax_rad': 3, 'num_bad_primes': 2, 'num_int_pts': 0, 'optimality': 1, 'potential_good_reduction': True, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[2, 2, 27], [3, 2, 32]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-1.2504050306721196109222355575', 'prec': 97}, 'szpiro_ratio': 3.1789709565273077, '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, 0, 0, 0, 2, 0, 3, 0, 0, 0, 6, 0, 3, 0, 0, 0, 2, 0, 3, 0], 'aplist': [0, 0, 2, 3, 6, 3, 2, 3, -6, -8, 0, -7, -8, 12, -6, 4, 6, 1, 3, -12, -15, 9, -12, 10, 9], 'class_deg': 1, 'class_size': 1, 'conductor': 1728, 'isogeny_matrix': [[1]], 'lmfdb_iso': '1728.x', 'trace_hash': 1312092152316179483}
• 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': '1.1732110160019687204198083455', 'prec': 97}, 'conductor': 1728, 'gens': [], 'heights': [], 'lmfdb_label': '1728.x1', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '2.6757106733798058445825143103', 'prec': 97}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 97}, 'special_value': {'__RealLiteral__': 0, 'data': '2.6757106733798058445825143103283817713', 'prec': 127}, 'tamagawa_product': 1, 'torsion_generators': [], 'xcoord_integral_points': []}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': 'a', '3': 'a', '5': [0, 0], '7': [0, 0], '11': [0, 0], '13': [0, 0], '17': [0, 0], '19': [0, 0], '23': [0, 0], '29': [0, 0], '31': [0, 0, 0], '37': [0, 0], '41': [0, 0], '43': [0, 0], '47': [0, 0]}, 'iwp0': 0, 'lmfdb_label': '1728.x1'}
• ec_torsion_growth •

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

  
  
  • id: 87889
    {'degree': 3, 'field': [-2, 0, 0, 1], 'lmfdb_label': '1728.x1', 'torsion': [2]}
  • id: 87890
    {'degree': 6, 'field': [1, -3, 0, 5, 0, -3, 1], 'lmfdb_label': '1728.x1', 'torsion': [2, 2]}
  • id: 87891
    {'degree': 8, 'field': [-3, 0, 0, 0, -6, 0, 0, 0, 1], 'lmfdb_label': '1728.x1', 'torsion': [3]}
  • id: 87892
    {'degree': 12, 'field': [-3, 0, 18, 0, 39, 0, 48, 0, 15, 0, 6, 0, 1], 'lmfdb_label': '1728.x1', 'torsion': [4]}
  • id: 87893
    {'degree': 16, 'field': [9, 0, 0, 0, -18, 0, 0, 0, 39, 0, 0, 0, 6, 0, 0, 0, 1], 'lmfdb_label': '1728.x1', 'torsion': [3, 3]}
• 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: 27705
    {'conductor': 1728, 'conductor_valuation': 6, 'discriminant_valuation': 6, 'j_denominator_valuation': 0, 'kodaira_symbol': 2, 'lmfdb_label': '1728.x1', 'prime': 2, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 1}
  • id: 27706
    {'conductor': 1728, 'conductor_valuation': 3, 'discriminant_valuation': 9, 'j_denominator_valuation': 0, 'kodaira_symbol': -4, 'lmfdb_label': '1728.x1', 'prime': 3, '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: 1675372
    {'adelic_gens': [[14, 19, 17, 14], [13, 12, 12, 13], [7, 0, 6, 7], [3, 4, 8, 3], [4, 9, 3, 7], [11, 0, 0, 23], [1, 0, 12, 1], [5, 0, 0, 5], [23, 18, 18, 23], [1, 12, 0, 1], [19, 19, 13, 14]], 'adelic_image': '24.24.1.?', 'conductor': 1728, 'lmfdb_label': '1728.x1', 'prime': 0}
  • id: 8503
    {'conductor': 1728, 'elladic_image': '3.3.0.1', 'lmfdb_label': '1728.x1', 'modell_image': '3Nn', 'prime': 3}
• lmfdb_iso 1728.x does not appear in ec_padic