Elliptic curve data - 1728.o4

Formats: - HTML - YAML - JSON - 2026-05-05T17:59:40.749394

• 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': '1728v', 'Clabel': '1728v3', 'Cnumber': 3, 'absD': 1259712, 'ainvs': [0, 0, 0, 0, 54], 'analytic_rank': 1, 'bad_primes': [2, 3], 'class_deg': 27, 'class_size': 4, 'cm': -3, 'conductor': 1728, 'degree': 144, 'elladic_images': ['27.972.55.16'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.15058462164698299173481920743', 'prec': 107}, 'faltings_index': 1, 'faltings_ratio': 3, 'intrinsic_torsion': 1, 'iso_nlabel': 14, 'isogeny_degrees': [1, 3, 9], 'jinv': [0, 1], 'lmfdb_iso': '1728.o', 'lmfdb_label': '1728.o4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, 0, 1, 0, 0], 'min_quad_twist_disc': 24, 'modell_images': ['3Cs'], 'nonmax_primes': [3], 'nonmax_rad': 3, 'num_bad_primes': 2, 'num_int_pts': 2, 'optimality': 0, 'potential_good_reduction': True, 'rank': 1, 'regulator': {'__RealLiteral__': 0, 'data': '0.44985894522670601515925372855', 'prec': 103}, 'semistable': False, 'serre_invariants': [[2, 2, 27], [3, 2, 32]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-1.3211174284280379149898691959', 'prec': 103}, 'szpiro_ratio': 2.8842282173954805, '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, 0, 0, 1, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -7, 0], 'aplist': [0, 0, 0, 1, 0, -5, 0, -7, 0, 0, 4, -11, 0, 8, 0, 0, 0, 1, 5, 0, -7, -17, 0, 0, -19], 'class_deg': 27, 'class_size': 4, 'conductor': 1728, 'isogeny_matrix': [[1, 27, 9, 3], [27, 1, 3, 9], [9, 3, 1, 3], [3, 9, 3, 1]], 'lmfdb_iso': '1728.o', 'trace_hash': 1415855867502297635}
• 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': '1.3514380440104491068077171701', 'prec': 100}, 'conductor': 1728, 'gens': [[3, 9, 1]], 'heights': [{'__RealLiteral__': 0, 'data': '0.44985894522670601515925372855', 'prec': 103}], 'lmfdb_label': '1728.o4', 'ngens': 1, 'rank_bounds': [1, 1], 'real_period': {'__RealLiteral__': 0, 'data': '2.1636817490137510094722028257', 'prec': 100}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '2.9200547682528014610166866824285120714', 'prec': 130}, 'tamagawa_product': 3, 'torsion_generators': [], 'xcoord_integral_points': [3]}
• ec_iwasawa •

  Column	Type
  iwdata	jsonb
  iwp0	smallint
  lmfdb_label	text

  
  {'iwdata': {'2': 'a', '3': 'a', '5': [1, 1, 0], '7': [5, 0], '11': [1, 1, 0], '13': [1, 0], '17': [1, 1, 0], '19': [3, 0], '23': [1, 1, 0], '29': [1, 1, 0], '31': [1, 0], '37': [1, 0], '41': [1, 1, 0], '43': [1, 0], '47': [1, 1, 0]}, 'iwp0': 0, 'lmfdb_label': '1728.o4'}
• ec_torsion_growth •

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

  
  
  • id: 87972
    {'degree': 2, 'field': [-6, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [3]}
  • id: 87973
    {'degree': 2, 'field': [2, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [3]}
  • id: 87974
    {'degree': 3, 'field': [-2, 0, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [2]}
  • id: 87975
    {'degree': 4, 'field': [4, 0, -2, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [3, 3]}
  • id: 87977
    {'degree': 6, 'field': [-24, 0, 36, 0, -12, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [9]}
  • id: 87978
    {'degree': 6, 'field': [-6, 0, 12, 0, -6, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [6]}
  • id: 87976
    {'degree': 6, 'field': [1, -3, 0, 5, 0, -3, 1], 'lmfdb_label': '1728.o4', 'torsion': [2, 2]}
  • id: 87979
    {'degree': 6, 'field': [2, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [6]}
  • id: 87980
    {'degree': 12, 'field': [-3, 0, 0, 0, 3, 0, 0, 0, -9, 0, 0, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [4]}
  • id: 87983
    {'degree': 12, 'field': [4, 0, -24, 0, 60, 0, -44, 0, 0, 0, 6, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [6, 6]}
  • id: 87981
    {'degree': 12, 'field': [64, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [3, 9]}
  • id: 87982
    {'degree': 12, 'field': [1344, 0, 2016, 0, 336, 0, -480, 0, 72, 0, -6, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [7]}
  • id: 87985
    {'degree': 18, 'field': [-24, 0, 144, 0, -360, 0, 516, 0, -432, 0, 180, 0, -12, 0, -12, 0, 0, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [18]}
  • id: 87984
    {'degree': 18, 'field': [512, 0, 0, 0, 0, 0, 1920, 0, 0, 0, 0, 0, 168, 0, 0, 0, 0, 0, 1], 'lmfdb_label': '1728.o4', 'torsion': [9]}
• 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: 27723
    {'conductor': 1728, 'conductor_valuation': 6, 'discriminant_valuation': 6, 'j_denominator_valuation': 0, 'kodaira_symbol': 2, 'lmfdb_label': '1728.o4', 'prime': 2, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 1}
  • id: 27724
    {'conductor': 1728, 'conductor_valuation': 3, 'discriminant_valuation': 9, 'j_denominator_valuation': 0, 'kodaira_symbol': -4, 'lmfdb_label': '1728.o4', 'prime': 3, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 3}
• ec_galrep •

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

  
  {'conductor': 1728, 'elladic_image': '27.972.55.16', 'lmfdb_label': '1728.o4', 'modell_image': '3Cs', 'prime': 3}
• ec_padic •

  Column	Type
  lmfdb_iso	text
  p	smallint
  prec	smallint
  unit	numeric
  val	smallint

  
  
  • id: 47970
    {'lmfdb_iso': '1728.o', 'p': 7, 'prec': 20, 'unit': 64466393800564870, 'val': 0}
  • id: 47973
    {'lmfdb_iso': '1728.o', 'p': 13, 'prec': 20, 'unit': 238153760673429703495, 'val': 1}
  • id: 47975
    {'lmfdb_iso': '1728.o', 'p': 19, 'prec': 20, 'unit': 94004648432768508193260, 'val': 2}
  • id: 47977
    {'lmfdb_iso': '1728.o', 'p': 31, 'prec': 20, 'unit': 12619005292858312548951081581, 'val': 1}
  • id: 47969
    {'lmfdb_iso': '1728.o', 'p': 37, 'prec': 20, 'unit': 517362074490160783924396477122, 'val': 1}
  • id: 47972
    {'lmfdb_iso': '1728.o', 'p': 43, 'prec': 20, 'unit': 3087603812699019464049631429247, 'val': 1}
  • id: 47976
    {'lmfdb_iso': '1728.o', 'p': 61, 'prec': 20, 'unit': 151259172822003095902853785112513908, 'val': 0}
  • id: 47968
    {'lmfdb_iso': '1728.o', 'p': 67, 'prec': 20, 'unit': 44232311809664014036836424880869767, 'val': 1}
  • id: 47971
    {'lmfdb_iso': '1728.o', 'p': 73, 'prec': 20, 'unit': 9906620409695149067343618220492485, 'val': 1}
  • id: 47974
    {'lmfdb_iso': '1728.o', 'p': 79, 'prec': 20, 'unit': 520945465242566161265436488804170554, 'val': 1}
  • id: 47967
    {'lmfdb_iso': '1728.o', 'p': 97, 'prec': 20, 'unit': 36055190393614427169637265108839436338, 'val': 1}