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ec_curvedata • Show schema
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{'Ciso': '21a', 'Clabel': '21a6', 'Cnumber': 6, 'abc_quality': 1.042661241253942, 'absD': 17294403, 'adelic_genus': 1, 'adelic_index': 192, 'adelic_level': 336, 'ainvs': [1, 0, 0, -34, -217], 'analytic_rank': 0, 'bad_primes': [3, 7], 'class_deg': 8, 'class_size': 6, 'cm': 0, 'conductor': 21, 'degree': 4, 'elladic_images': ['8.48.0.177'], 'faltings_height': {'__RealLiteral__': 0, 'data': '0.074205383390939770758265854672', 'prec': 107}, 'faltings_index': 4, 'faltings_ratio': 8, 'iso_nlabel': 0, 'isogeny_degrees': [1, 2, 4, 8], 'jinv': [-4354703137, 17294403], 'lmfdb_iso': '21.a', 'lmfdb_label': '21.a4', 'lmfdb_number': 4, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, 0, 0, -34, -217], 'min_quad_twist_disc': 1, 'modell_images': ['2B'], 'modm_images': ['2.3.0.a.1', '4.12.0-4.c.1.2', '6.6.0.a.1', '8.48.0-8.bb.2.3', '12.24.0-12.g.1.1', '24.96.0-24.bj.1.1', '112.96.0.?', '336.192.1.?'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 2, 'num_int_pts': 0, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': True, 'serre_invariants': [[2, 2, 3], [3, 4, 7], [7, 8, 3]], 'sha': 1, 'sha_primes': [], 'signD': -1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '0.074205383390939770758265854672', 'prec': 107}, 'szpiro_ratio': 7.967306624104166, 'torsion': 2, 'torsion_primes': [2], 'torsion_structure': [2]}
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ec_classdata • Show schema
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{'anlist': [0, 1, -1, 1, -1, -2, -1, -1, 3, 1, 2, 4, -1, -2, 1, -2, -1, -6, -1, 4, 2], 'aplist': [-1, 1, -2, -1, 4, -2, -6, 4, 0, -2, 0, 6, 2, -4, 0, 6, 12, -2, 4, 0, -6, -16, -12, -14, 18], 'class_deg': 8, 'class_size': 6, 'conductor': 21, 'isogeny_matrix': [[1, 2, 8, 4, 4, 8], [2, 1, 4, 2, 2, 4], [8, 4, 1, 8, 2, 4], [4, 2, 8, 1, 4, 8], [4, 2, 2, 4, 1, 2], [8, 4, 4, 8, 2, 1]], 'lmfdb_iso': '21.a', 'trace_hash': 1161919666425559947}
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ec_mwbsd • Show schema
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{'area': {'__RealLiteral__': 0, 'data': '0.86207692963712698063615726709', 'prec': 103}, 'conductor': 21, 'gens': [], 'heights': [], 'lmfdb_label': '21.a4', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '0.90223081077698411118680031281', 'prec': 103}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.45111540538849205559340015640263020881', 'prec': 133}, 'tamagawa_product': 2, 'torsion_generators': [[31, -31, 2]], 'xcoord_integral_points': []}
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ec_iwasawa • Show schema
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{'iwdata': {'2': [1, 2], '3': [1, 0], '7': [0, 0]}, 'iwp0': 3, 'lmfdb_label': '21.a4'}
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ec_torsion_growth • Show schema
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id: 472
{'degree': 2, 'field': [-3, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [4]}
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id: 471
{'degree': 2, 'field': [1, -1, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 2]}
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id: 473
{'degree': 2, 'field': [1, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [4]}
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id: 476
{'degree': 4, 'field': [-3, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [8]}
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id: 474
{'degree': 4, 'field': [1, 0, -1, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 4]}
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id: 478
{'degree': 4, 'field': [1, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [8]}
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id: 475
{'degree': 4, 'field': [3, 0, -3, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 4]}
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id: 477
{'degree': 4, 'field': [9, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [8]}
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id: 483
{'degree': 8, 'field': [-3, 0, -1, 0, -6, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [6]}
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id: 479
{'degree': 8, 'field': [1, -4, 8, -10, 7, -2, 2, -2, 1], 'lmfdb_label': '21.a4', 'torsion': [4, 4]}
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id: 480
{'degree': 8, 'field': [1, 0, 0, 0, -1, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 8]}
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id: 482
{'degree': 8, 'field': [3, 0, 0, 0, -3, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 8]}
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id: 481
{'degree': 8, 'field': [9, 0, 0, 0, 3, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 8]}
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id: 484
{'degree': 16, 'field': [1, -12, 68, -232, 488, -564, 194, 396, -665, 444, -62, -152, 162, -92, 34, -8, 1], 'lmfdb_label': '21.a4', 'torsion': [4, 8]}
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id: 490
{'degree': 16, 'field': [9, 0, -9, 0, -99, 0, -285, 0, 177, 0, -24, 0, -14, 0, 5, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [12]}
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id: 489
{'degree': 16, 'field': [9, 0, -3, 0, -17, 0, -6, 0, 39, 0, -2, 0, 6, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [2, 6]}
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id: 485
{'degree': 16, 'field': [9, 0, 0, 0, 54, 0, 0, 0, 87, 0, 0, 0, 6, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [4, 8]}
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id: 486
{'degree': 16, 'field': [9, 0, 0, 0, 360, 0, 0, 0, 507, 0, 0, 0, -120, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [16]}
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id: 491
{'degree': 16, 'field': [9, 0, 72, 0, 144, 0, 31, 0, 80, 0, -23, 0, 1, 0, 2, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [12]}
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id: 488
{'degree': 16, 'field': [1296, 0, 0, 0, -864, 0, 0, 0, 265, 0, 0, 0, -24, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [16]}
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id: 487
{'degree': 16, 'field': [104976, 0, 0, 0, -23328, 0, 0, 0, 2385, 0, 0, 0, -72, 0, 0, 0, 1], 'lmfdb_label': '21.a4', 'torsion': [16]}
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ec_localdata • Show schema
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id: 56
{'conductor': 21, 'conductor_valuation': 1, 'discriminant_valuation': 1, 'j_denominator_valuation': 1, 'kodaira_symbol': 5, 'lmfdb_label': '21.a4', 'prime': 3, 'reduction_type': 1, 'root_number': -1, 'tamagawa_number': 1}
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id: 57
{'conductor': 21, 'conductor_valuation': 1, 'discriminant_valuation': 8, 'j_denominator_valuation': 8, 'kodaira_symbol': 12, 'lmfdb_label': '21.a4', 'prime': 7, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}
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ec_galrep • Show schema
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id: 1665919
{'adelic_gens': [[5, 4, 332, 333], [241, 16, 248, 129], [1, 16, 0, 1], [232, 1, 191, 10], [15, 2, 238, 323], [55, 16, 54, 73], [321, 16, 320, 17], [1, 0, 16, 1], [13, 16, 164, 105]], 'adelic_image': '336.192.1.?', 'conductor': 21, 'lmfdb_label': '21.a4', 'prime': 0}
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id: 41
{'conductor': 21, 'elladic_image': '8.48.0.177', 'lmfdb_label': '21.a4', 'modell_image': '2B', 'prime': 2}
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lmfdb_iso 21.a does not appear in ec_padic