Elliptic curve data - 15.a2

Formats: - HTML - YAML - JSON - 2025-12-16T05:56:45.680310

• 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': '15a', 'Clabel': '15a2', 'Cnumber': 2, 'abc_quality': 1.038972011651739, 'absD': 164025, 'adelic_genus': 13, 'adelic_index': 768, 'adelic_level': 240, 'ainvs': [1, 1, 1, -135, -660], 'analytic_rank': 0, 'bad_primes': [3, 5], 'class_deg': 16, 'class_size': 8, 'cm': 0, 'conductor': 15, 'degree': 2, 'elladic_images': ['8.96.0.58'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.055703981802871132734980436436', 'prec': 110}, 'faltings_index': 4, 'faltings_ratio': 8, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4, 8], 'jinv': [272223782641, 164025], 'lmfdb_iso': '15.a', 'lmfdb_label': '15.a2', 'lmfdb_number': 2, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, 1, 1, -135, -660], 'min_quad_twist_disc': 1, 'modell_images': ['2Cs'], 'modm_images': ['2.6.0.a.1', '4.24.0-4.b.1.1', '8.96.0-8.k.1.1', '20.48.0-20.c.1.2', '40.192.1-40.cc.2.4', '48.192.1-48.e.2.2', '80.384.5.?', '240.768.13.?'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 2, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': True, 'serre_invariants': [[2, 2, 1], [3, 4, 5], [5, 6, 3]], 'sha': 1, 'sha_primes': [], 'signD': 1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-0.055703981802871132734980436436', 'prec': 110}, 'szpiro_ratio': 9.72282208231094, '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, -1, -1, -1, 1, 1, 0, 3, 1, -1, -4, 1, -2, 0, -1, -1, 2, -1, 4, -1], 'aplist': [-1, -1, 1, 0, -4, -2, 2, 4, 0, -2, 0, -10, 10, 4, 8, -10, -4, -2, 12, -8, 10, 0, 12, -6, 2], 'class_deg': 16, 'class_size': 8, 'conductor': 15, 'isogeny_matrix': [[1, 2, 4, 16, 4, 8, 16, 8], [2, 1, 2, 8, 2, 4, 8, 4], [4, 2, 1, 16, 4, 8, 16, 8], [16, 8, 16, 1, 4, 2, 4, 8], [4, 2, 4, 4, 1, 2, 4, 2], [8, 4, 8, 2, 2, 1, 2, 4], [16, 8, 16, 4, 4, 2, 1, 8], [8, 4, 8, 8, 2, 4, 8, 1]], 'lmfdb_iso': '15.a', 'trace_hash': 750292609497685491}
• 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': '1.1178508563087645508833713911', 'prec': 100}, 'conductor': 15, 'gens': [], 'heights': [], 'lmfdb_label': '15.a2', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '1.4006030423326020231801808368', 'prec': 100}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.35015076058315050579504520920239987269', 'prec': 133}, 'tamagawa_product': 4, 'torsion_generators': [[-7, 3, 1], [13, -7, 1]], 'xcoord_integral_points': [-7, 13]}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': [0, 2], '3': [0, 0], '5': [1, 0]}, 'iwp0': 3, 'lmfdb_label': '15.a2'}
• ec_torsion_growth •

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

  
  
  • id: 133
    {'degree': 2, 'field': [-1, -1, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 4]}
  • id: 134
    {'degree': 2, 'field': [1, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 4]}
  • id: 135
    {'degree': 2, 'field': [5, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 4]}
  • id: 139
    {'degree': 4, 'field': [-5, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 8]}
  • id: 137
    {'degree': 4, 'field': [1, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 8]}
  • id: 136
    {'degree': 4, 'field': [1, 0, 3, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [4, 4]}
  • id: 138
    {'degree': 4, 'field': [25, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 8]}
  • id: 143
    {'degree': 8, 'field': [-1, 4, -4, -1, 4, -7, 7, -4, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 6]}
  • id: 142
    {'degree': 8, 'field': [1, 0, 0, 0, 7, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [4, 8]}
  • id: 141
    {'degree': 8, 'field': [1, 2, 2, -10, 11, -10, 8, -4, 1], 'lmfdb_label': '15.a2', 'torsion': [4, 8]}
  • id: 140
    {'degree': 8, 'field': [36, 0, -56, 0, 16, 0, 4, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [4, 8]}
  • id: 144
    {'degree': 16, 'field': [1, 0, 0, 0, -36, 0, 0, 0, 646, 0, 0, 0, -36, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [8, 8]}
  • id: 150
    {'degree': 16, 'field': [1, 0, 8, 0, 16, 0, -9, 0, -24, 0, 9, 0, 1, 0, 2, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 12]}
  • id: 151
    {'degree': 16, 'field': [9, 0, -63, 0, 135, 0, -146, 0, 125, 0, -89, 0, 43, 0, -10, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 12]}
  • id: 148
    {'degree': 16, 'field': [25, 0, 0, 0, 350, 0, 0, 0, -360, 0, 0, 0, -25, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 16]}
  • id: 145
    {'degree': 16, 'field': [625, 0, 0, 0, -200, 0, 0, 0, 39, 0, 0, 0, -8, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 16]}
  • id: 149
    {'degree': 16, 'field': [625, 0, 1000, 0, 400, 0, 1875, 0, 600, 0, 165, 0, 25, 0, 10, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 12]}
  • id: 147
    {'degree': 16, 'field': [10000, 0, 0, 0, -4000, 0, 0, 0, 6225, 0, 0, 0, -40, 0, 0, 0, 1], 'lmfdb_label': '15.a2', 'torsion': [2, 16]}
  • id: 146
    {'degree': 16, 'field': [10000, 0, 0, 0, 4000, 0, 0, 0, 6225, 0, 0, 0, 40, 0, 0, 0, 1], 'lmfdb_label': '15.a2', '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: 17
    {'conductor': 15, 'conductor_valuation': 1, 'discriminant_valuation': 8, 'j_denominator_valuation': 8, 'kodaira_symbol': 12, 'lmfdb_label': '15.a2', 'prime': 3, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}
  • id: 18
    {'conductor': 15, 'conductor_valuation': 1, 'discriminant_valuation': 2, 'j_denominator_valuation': 2, 'kodaira_symbol': 6, 'lmfdb_label': '15.a2', 'prime': 5, '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: 1665898
    {'adelic_gens': [[1, 0, 16, 1], [1, 16, 0, 1], [9, 8, 178, 215], [161, 8, 0, 1], [61, 8, 170, 197], [1, 16, 4, 65], [1, 16, 0, 181], [225, 16, 224, 17]], 'adelic_image': '240.768.13.?', 'conductor': 15, 'lmfdb_label': '15.a2', 'prime': 0}
  • id: 16
    {'conductor': 15, 'elladic_image': '8.96.0.58', 'lmfdb_label': '15.a2', 'modell_image': '2Cs', 'prime': 2}
• lmfdb_iso 15.a does not appear in ec_padic