LMFDB - Elliptic curve data

Formats: - HTML - YAML - JSON - 2025-10-21T15:36:26.767368

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[]

ec_curvedata • Show schema

{'Ciso': '21840x', 'Clabel': '21840x2', 'Cnumber': 2, 'abc_quality': 1.023218976193033, 'absD': 132072351989760000000000, 'adelic_genus': 0, 'adelic_index': 48, 'adelic_level': 840, 'ainvs': [0, -1, 0, -26066496, -48138568704], 'analytic_rank': 0, 'bad_primes': [2, 3, 5, 7, 13], 'class_deg': 4, 'class_size': 4, 'cm': 0, 'conductor': 21840, 'degree': 2211840, 'elladic_images': ['8.12.0.1'], 'faltings_height': {'__RealLiteral__': 0, 'data': '3.1852974550065941378832392695', 'prec': 100}, 'faltings_index': 1, 'faltings_ratio': 2, 'iso_nlabel': 1, 'isogeny_degrees': [1, 2], 'jinv': [478202393398338853167169, 32244226560000000000], 'lmfdb_iso': '21840.b', 'lmfdb_label': '21840.b2', 'lmfdb_number': 2, 'manin_constant': 1, 'min_quad_twist_ainvs': [1, 0, 0, -1629156, 752165136], 'min_quad_twist_disc': -4, 'modell_images': ['2Cs'], 'modm_images': ['2.6.0.a.1', '8.12.0-2.a.1.1', '28.12.0-2.a.1.1', '56.24.0-56.a.1.2', '60.12.0-2.a.1.1', '120.24.0.?', '420.24.0.?', '840.48.0.?'], 'nonmax_primes': [2], 'nonmax_rad': 2, 'num_bad_primes': 5, 'num_int_pts': 3, 'optimality': 0, 'potential_good_reduction': False, 'rank': 0, 'regulator': 1, 'semistable': False, 'serre_invariants': [[2, 2, 1], [3, 4, 7280], [5, 6, 4368], [7, 8, 3120], [13, 14, 1680]], 'sha': 1, 'sha_primes': [], 'signD': 1, 'stable_faltings_height': {'__RealLiteral__': 0, 'data': '2.4921502744466488284660071480', 'prec': 100}, 'szpiro_ratio': 6.289555827964953, 'torsion': 4, 'torsion_primes': [2], 'torsion_structure': [2, 2]}

Column	Type
anlist	smallint[]
aplist	smallint[]
class_deg	smallint
class_size	smallint
conductor	integer
isogeny_matrix	smallint[]
lmfdb_iso	text
trace_hash	bigint

ec_classdata • Show schema

{'anlist': [0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 2, 0, -4, 0], 'aplist': [0, -1, -1, -1, 0, -1, 2, -4, -8, -6, -4, -6, -10, 0, -12, -2, -4, 10, -8, -8, 6, 0, 4, 6, 6], 'class_deg': 4, 'class_size': 4, 'conductor': 21840, 'isogeny_matrix': [[1, 2, 4, 4], [2, 1, 2, 2], [4, 2, 1, 4], [4, 2, 4, 1]], 'lmfdb_iso': '21840.b', 'trace_hash': 95396581099167074}

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[]

ec_mwbsd • Show schema

{'area': {'__RealLiteral__': 0, 'data': '0.0017111409478199071207164733178', 'prec': 110}, 'conductor': 21840, 'gens': [], 'heights': [], 'lmfdb_label': '21840.b2', 'ngens': 0, 'rank_bounds': [0, 0], 'real_period': {'__RealLiteral__': 0, 'data': '0.067099009618601796980656635806', 'prec': 107}, 'sha_an': {'__RealLiteral__': 0, 'data': '1.0000000000000000000000000000', 'prec': 100}, 'special_value': {'__RealLiteral__': 0, 'data': '0.26839603847440718792262654322563945887', 'prec': 133}, 'tamagawa_product': 64, 'torsion_generators': [[-2336, 0, 1], [5856, 0, 1]], 'xcoord_integral_points': [-3519, -2336, 5856]}

Column	Type
iwdata	jsonb
iwp0	smallint
lmfdb_label	text

ec_iwasawa • Show schema
{'iwdata': {'2': 'a', '3': [0, 0], '5': [0, 0], '7': [0, 0], '13': [0, 0]}, 'iwp0': 3, 'lmfdb_label': '21840.b2'}

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

ec_torsion_growth • Show schema

id: 1109378
{'degree': 4, 'field': [4, 0, 11, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 4]}
id: 1109377
{'degree': 4, 'field': [9, 0, 8, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 4]}
id: 1109379
{'degree': 4, 'field': [49, 0, -16, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 4]}
id: 1109380
{'degree': 8, 'field': [-48, 0, 181, 0, -24, 0, 0, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 6]}
id: 1109384
{'degree': 16, 'field': [14161, 0, -34344, 0, 32800, 0, -16536, 0, 4914, 0, -936, 0, 160, 0, -24, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 8]}
id: 1109381
{'degree': 16, 'field': [67104, 169920, 51360, -60288, 76402, 48084, -37930, -16152, 13881, 840, -2630, 72, 360, -12, -28, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [4, 4]}
id: 1109382
{'degree': 16, 'field': [212722225, 0, -118448392, 0, 38238496, 0, 787464, 0, 46098, 0, -3336, 0, 928, 0, 8, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 8]}
id: 1109383
{'degree': 16, 'field': [14086366596, 0, -2143934280, 0, 357480069, 0, -32103390, 0, 2453611, 0, -135020, 0, 5259, 0, -110, 0, 1], 'lmfdb_label': '21840.b2', 'torsion': [2, 8]}

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

ec_localdata • Show schema

id: 497969
{'conductor': 21840, 'conductor_valuation': 4, 'discriminant_valuation': 30, 'j_denominator_valuation': 18, 'kodaira_symbol': -26, 'lmfdb_label': '21840.b2', 'prime': 2, 'reduction_type': 0, 'root_number': -1, 'tamagawa_number': 4}
id: 497970
{'conductor': 21840, 'conductor_valuation': 1, 'discriminant_valuation': 2, 'j_denominator_valuation': 2, 'kodaira_symbol': 6, 'lmfdb_label': '21840.b2', 'prime': 3, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}
id: 497971
{'conductor': 21840, 'conductor_valuation': 1, 'discriminant_valuation': 10, 'j_denominator_valuation': 10, 'kodaira_symbol': 14, 'lmfdb_label': '21840.b2', 'prime': 5, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}
id: 497972
{'conductor': 21840, 'conductor_valuation': 1, 'discriminant_valuation': 2, 'j_denominator_valuation': 2, 'kodaira_symbol': 6, 'lmfdb_label': '21840.b2', 'prime': 7, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}
id: 497973
{'conductor': 21840, 'conductor_valuation': 1, 'discriminant_valuation': 4, 'j_denominator_valuation': 4, 'kodaira_symbol': 8, 'lmfdb_label': '21840.b2', 'prime': 13, 'reduction_type': -1, 'root_number': 1, 'tamagawa_number': 2}

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

ec_galrep • Show schema

id: 1809993
{'adelic_gens': [[837, 4, 836, 5], [1, 4, 0, 1], [421, 4, 2, 9], [241, 2, 0, 1], [3, 2, 278, 839], [171, 2, 166, 839], [1, 0, 4, 1], [627, 836, 418, 837]], 'adelic_image': '840.48.0.?', 'conductor': 21840, 'lmfdb_label': '21840.b2', 'prime': 0}
id: 100906
{'conductor': 21840, 'elladic_image': '8.12.0.1', 'lmfdb_label': '21840.b2', 'modell_image': '2Cs', 'prime': 2}

lmfdb_iso 21840.b does not appear in ec_padic

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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
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Abelian varieties over $\F_{q}$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.