Formats: - HTML - YAML - JSON - 2026-08-24T23:57:07.066293
  • ec_curvedata • Show schema
    {'Ciso': '48a', 'Clabel': '48a1', 'Cnumber': 1, 'abc_quality': 0.972547111469975, 'absD': 2304, 'adelic_genus': 1, 'adelic_index': 192, 'adelic_level': 24, 'ainvs': [0, 1, 0, -4, -4], 'analytic_rank': 0, 'bad_primes': [2, 3], 'class_deg': 8, 'class_size': 6, 'cm': 0, 'conductor': 48, 'degree': 2, 'elladic_images': ['8.96.0.53'], 'faltings_height': {'__RealLiteral__': 0, 'data': '-0.64535228607985727265901067683', 'prec': 97}, 'faltings_index': 1, 'faltings_ratio': 2, 'intrinsic_torsion': 2, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4], 'jinv': [35152, 9], 'lmfdb_iso': '48.a', 'lmfdb_label': '48.a4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [0, -1, 0, -4, 4], 'min_quad_twist_disc': -4, 'modell_images': ['2Cs'], 'modm_images': ['2.6.0.a.1', '4.24.0-4.b.1.1', '8.96.0-8.h.1.2', '12.48.0-12.c.1.3', '24.192.1-24.bu.1.1'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 3, 'optimality': 1, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[2, 2, 1], [3, 4, 16]], 'sha': 1, 'sha_primes': [], 'signD': 1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '-1.1074504064531541456038320911', 'prec': 97}, 'szpiro_ratio': 4.136342750992568, 'torsion': 4, 'torsion_primes': [2], 'torsion_structure': [2, 2]}
  • ec_classdata • Show schema
    {'anlist': [0, 1, 0, 1, 0, -2, 0, 0, 0, 1, 0, -4, 0, -2, 0, -2, 0, 2, 0, 4, 0], 'aplist': [0, 1, -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': 48, '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': '48.a', 'trace_hash': 1588924079463638771}
  • ec_mwbsd • Show schema
    {'area': {'__RealLiteral__': 0, 'data': '3.6353470179314388805624933342', 'prec': 97}, 'conductor': 48, 'gens': [], 'heights': [], 'lmfdb_label': '48.a4', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '3.3715007096251920857424073156', 'prec': 97}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 97}, 'special_value': {'__RealLiteral__': 0, 'data': '0.84287517740629802143560182889957071428', 'prec': 127}, 'tamagawa_product': 4, 'torsion_generators': [[-1, 0, 1], [2, 0, 1]], 'xcoord_integral_points': [-2, -1, 2]}
  • ec_iwasawa • Show schema
    {'iwdata': {'2': 'a', '3': [1, 0]}, 'iwp0': 3, 'lmfdb_label': '48.a4'}
  • ec_torsion_growth • Show schema
    • id: 1515
      {'degree': 2, 'field': [-3, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 4]}
    • id: 1513
      {'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 4]}
    • id: 1514
      {'degree': 2, 'field': [1, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 4]}
    • id: 1517
      {'degree': 4, 'field': [1, -2, 0, -2, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 8]}
    • id: 1516
      {'degree': 4, 'field': [1, 0, -1, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [4, 4]}
    • id: 1518
      {'degree': 4, 'field': [1, 0, 0, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 8]}
    • id: 1522
      {'degree': 8, 'field': [-3, 0, -4, 0, -6, 0, 0, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 6]}
    • id: 1521
      {'degree': 8, 'field': [1, -4, 8, -10, 7, -2, 2, -2, 1], 'lmfdb_label': '48.a4', 'torsion': [4, 8]}
    • id: 1519
      {'degree': 8, 'field': [1, 0, -4, 0, 3, 0, 2, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 8]}
    • id: 1520
      {'degree': 8, 'field': [1, 0, 0, 0, -1, 0, 0, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [4, 8]}
    • id: 1527
      {'degree': 16, 'field': [-2, 16, -40, 0, 158, -220, -34, 280, -138, -116, 124, -10, -30, 14, 2, -4, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 12]}
    • id: 1523
      {'degree': 16, 'field': [1, -12, 68, -232, 488, -564, 194, 396, -665, 444, -62, -152, 162, -92, 34, -8, 1], 'lmfdb_label': '48.a4', 'torsion': [8, 8]}
    • id: 1528
      {'degree': 16, 'field': [1, -4, 6, -10, 28, -30, 24, -36, 60, -36, 24, -30, 28, -10, 6, -4, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 12]}
    • id: 1524
      {'degree': 16, 'field': [4, 0, -32, 0, 64, 0, -80, 0, 76, 0, -56, 0, 28, 0, -8, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 16]}
    • id: 1525
      {'degree': 16, 'field': [4, 0, -16, 0, 40, 0, 0, 0, -24, 0, -8, 0, 20, 0, -8, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 16]}
    • id: 1526
      {'degree': 16, 'field': [9, 0, -12, 0, -2, 0, -24, 0, 39, 0, -8, 0, 6, 0, 0, 0, 1], 'lmfdb_label': '48.a4', 'torsion': [2, 12]}
  • ec_localdata • Show schema
    • id: 215
      {'conductor': 48, 'conductor_valuation': 4, 'discriminant_valuation': 8, 'j_denominator_valuation': 0, 'kodaira_symbol': -1, 'lmfdb_label': '48.a4', 'prime': 2, 'reduction_type': 0, 'root_number': 1, 'tamagawa_number': 2}
    • id: 216
      {'conductor': 48, 'conductor_valuation': 1, 'discriminant_valuation': 2, 'j_denominator_valuation': 2, 'kodaira_symbol': 6, 'lmfdb_label': '48.a4', 'prime': 3, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 2}
  • ec_galrep • Show schema
    • id: 1665991
      {'adelic_gens': [[1, 0, 8, 1], [1, 8, 0, 1], [17, 8, 16, 9], [23, 2, 6, 19], [5, 4, 20, 21], [17, 0, 16, 5], [19, 16, 2, 21]], 'adelic_image': '24.192.1.?', 'conductor': 48, 'lmfdb_label': '48.a4', 'prime': 0}
    • id: 129
      {'conductor': 48, 'elladic_image': '8.96.0.53', 'lmfdb_label': '48.a4', 'modell_image': '2Cs', 'prime': 2}
  • lmfdb_iso 48.a does not appear in ec_padic