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377774 bytes, as of the version from 2026-09-17 13:35 (current). Recorded here, not run.
"""Class representatives from John Cremona's ecnf-data.
Generated from ecnf-data commit 10b28418e80392032b106ea00e6c5aa109d28e7b. The table uses one representative
curve, the first curve in each LMFDB isogeny class, for every class over
D in {5, 8, 12, 13, 17} with conductor norm at most 150.
"""
SOURCE_COMMIT = '10b28418e80392032b106ea00e6c5aa109d28e7b'
MAX_CONDUCTOR_NORM = 150
RECORDS = [{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '31.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(5*w-2)',
'conductor_norm': 31,
'ainvs': '1,0;1,1;0,1;0,1;0,0',
'equation': '{y}^2+{x}{y}+\\phi{y}={x}^{3}+\\left(\\phi+1\\right){x}^{2}+\\phi{x}',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 8,
'omega': '51.508839712536626724604612731543070391',
'lvalue': '0.35992895949803944944002575466348575048',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-31.1-a1',
'class_label': '2.2.5.1-31.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '31.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(5*w-3)',
'conductor_norm': 31,
'ainvs': '1,0;-1,-1;0,1;-45,-30;-117,-111',
'equation': '{y}^2+{x}{y}+\\phi{y}={x}^{3}+\\left(-\\phi-1\\right){x}^{2}+\\left(-30\\phi-45\\right){x}-111\\phi-117',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '1.6096512410167695851438941478607209497',
'lvalue': '0.35992895949803944944002575466348575048',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-31.2-a1',
'class_label': '2.2.5.1-31.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '36.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(6)',
'conductor_norm': 36,
'ainvs': '1,1;0,1;0,1;0,0;0,0',
'equation': '{y}^2+\\left(\\phi+1\\right){x}{y}+\\phi{y}={x}^{3}+\\phi{x}^{2}',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 10,
'omega': '44.299621696875096561752925819698961425',
'lvalue': '0.39622786196694919547776387122068996572',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-36.1-a1',
'class_label': '2.2.5.1-36.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '41.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w+6)',
'conductor_norm': 41,
'ainvs': '0,0;0,-1;0,1;0,0;0,0',
'equation': '{y}^2+\\phi{y}={x}^{3}-\\phi{x}^{2}',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 7,
'omega': '46.260878468940992115535235203395846910',
'lvalue': '0.42221415900166715092967967717023093014',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-41.1-a1',
'class_label': '2.2.5.1-41.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '41.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w-7)',
'conductor_norm': 41,
'ainvs': '0,0;-1,1;1,1;-30,-10;-82,-32',
'equation': '{y}^2+\\left(\\phi+1\\right){y}={x}^{3}+\\left(\\phi-1\\right){x}^{2}+\\left(-10\\phi-30\\right){x}-32\\phi-82',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 1,
'omega': '0.94409956059063249215378031027338463082',
'lvalue': '0.42221415900166715092967967717023093014',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-41.2-a1',
'class_label': '2.2.5.1-41.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '45.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-6*w+3)',
'conductor_norm': 45,
'ainvs': '0,1;1,1;1,0;-7739,-4364;-296465,-255406',
'equation': '{y}^2+\\phi{x}{y}+{y}={x}^{3}+\\left(\\phi+1\\right){x}^{2}+\\left(-4364\\phi-7739\\right){x}-255406\\phi-296465',
'base_change': ['15.a3', '75.b3'],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '0.12260555513695878593710348088219652852',
'lvalue': '0.43864696912854141373237418324187134125',
'sha': 16,
'tamagawa_product': 2,
'label': '2.2.5.1-45.1-a1',
'class_label': '2.2.5.1-45.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '49.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(7)',
'conductor_norm': 49,
'ainvs': '0,0;1,-1;1,0;-29,-30;-84,-102',
'equation': '{y}^2+{y}={x}^{3}+\\left(-\\phi+1\\right){x}^{2}+\\left(-30\\phi-29\\right){x}-102\\phi-84',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 1,
'omega': '1.0454481928634487527701685488301141859',
'lvalue': '0.46753864523939638495454124542651620186',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-49.1-a1',
'class_label': '2.2.5.1-49.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '55.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w+7)',
'conductor_norm': 55,
'ainvs': '1,0;1,-1;1,0;-25,9;44,-6',
'equation': '{y}^2+{x}{y}+{y}={x}^{3}+\\left(-\\phi+1\\right){x}^{2}+\\left(9\\phi-25\\right){x}-6\\phi+44',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 6,
'omega': '19.867075744978321874989709057800649299',
'lvalue': '0.49360146533232004287759843238578959492',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-55.1-a1',
'class_label': '2.2.5.1-55.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '55.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-w+8)',
'conductor_norm': 55,
'ainvs': '0,1;1,-1;0,1;0,-1;0,0',
'equation': '{y}^2+\\phi{x}{y}+\\phi{y}={x}^{3}+\\left(-\\phi+1\\right){x}^{2}-\\phi{x}',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 6,
'omega': '39.734151489956643749979418115601298598',
'lvalue': '0.49360146533232004287759843238578959492',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-55.2-a1',
'class_label': '2.2.5.1-55.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '64.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(8)',
'conductor_norm': 64,
'ainvs': '0,0;-1,1;0,0;-25,14;-59,1',
'equation': '{y}^2={x}^{3}+\\left(\\phi-1\\right){x}^{2}+\\left(14\\phi-25\\right){x}+\\phi-59',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '2.3974174743597085652912019334743658850',
'lvalue': '0.53607884431141674535765569652422322021',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-64.1-a1',
'class_label': '2.2.5.1-64.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '71.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w+8)',
'conductor_norm': 71,
'ainvs': '0,1;1,1;0,1;-20,-4;-39,-37',
'equation': '{y}^2+\\phi{x}{y}+\\phi{y}={x}^{3}+\\left(\\phi+1\\right){x}^{2}+\\left(-4\\phi-20\\right){x}-37\\phi-39',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '2.4875763606722064105326074852261157155',
'lvalue': '0.55623898416845879826210782553394041678',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-71.1-a1',
'class_label': '2.2.5.1-71.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '71.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w-9)',
'conductor_norm': 71,
'ainvs': '1,1;-1,1;1,0;0,0;0,0',
'equation': '{y}^2+\\left(\\phi+1\\right){x}{y}+{y}={x}^{3}+\\left(\\phi-1\\right){x}^{2}',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 6,
'omega': '44.776374492099715389586934734070082880',
'lvalue': '0.55623898416845879826210782553394041678',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-71.2-a1',
'class_label': '2.2.5.1-71.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '76.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(8*w-6)',
'conductor_norm': 76,
'ainvs': '1,0;0,0;0,1;3,-29;-68,-28',
'equation': '{y}^2+{x}{y}+\\phi{y}={x}^{3}+\\left(-29\\phi+3\\right){x}-28\\phi-68',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 1,
'omega': '1.2944810943610766508855650560934576464',
'lvalue': '0.57890954451593741748915162110851663276',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-76.1-a1',
'class_label': '2.2.5.1-76.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '76.1',
'class': 'b',
'curve': 1,
'conductor_ideal': '(8*w-6)',
'conductor_norm': 76,
'ainvs': '1,1;0,0;1,0;-15986,3364;-793226,229016',
'equation': '{y}^2+\\left(\\phi+1\\right){x}{y}+{y}={x}^{3}+\\left(3364\\phi-15986\\right){x}+229016\\phi-793226',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 1,
'omega': '0.048505770308127107724947422977503923876',
'lvalue': '0.58569587843380091628719155570526876480',
'sha': 9,
'tamagawa_product': 3,
'label': '2.2.5.1-76.1-b1',
'class_label': '2.2.5.1-76.1-b'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '76.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-8*w+2)',
'conductor_norm': 76,
'ainvs': '1,1;0,-1;1,1;1,-3;0,-1',
'equation': '{y}^2+\\left(\\phi+1\\right){x}{y}+\\left(\\phi+1\\right){y}={x}^{3}-\\phi{x}^{2}+\\left(-3\\phi+1\\right){x}-\\phi',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 5,
'omega': '32.362027359026916272139126402336441162',
'lvalue': '0.57890954451593741748915162110851663276',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-76.2-a1',
'class_label': '2.2.5.1-76.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '76.2',
'class': 'b',
'curve': 1,
'conductor_ideal': '(-8*w+2)',
'conductor_norm': 76,
'ainvs': '0,1;1,-1;1,0;-12621,-3365;-564210,-229016',
'equation': '{y}^2+\\phi{x}{y}+{y}={x}^{3}+\\left(-\\phi+1\\right){x}^{2}+\\left(-3365\\phi-12621\\right){x}-229016\\phi-564210',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 1,
'omega': '0.048505770308127107724947422977503923876',
'lvalue': '0.58569587843380091628719155570526876480',
'sha': 9,
'tamagawa_product': 3,
'label': '2.2.5.1-76.2-b1',
'class_label': '2.2.5.1-76.2-b'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '79.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-8*w+5)',
'conductor_norm': 79,
'ainvs': '1,1;-1,1;0,1;0,0;0,0',
'equation': '{y}^2+\\left(\\phi+1\\right){x}{y}+\\phi{y}={x}^{3}+\\left(\\phi-1\\right){x}^{2}',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 4,
'omega': '21.557776634458581083298902274240298966',
'lvalue': '0.60255817498007528165769346017631645153',
'sha': 1,
'tamagawa_product': 1,
'label': '2.2.5.1-79.1-a1',
'class_label': '2.2.5.1-79.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '79.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-8*w+3)',
'conductor_norm': 79,
'ainvs': '0,1;1,1;0,0;-19,6;-43,4',
'equation': '{y}^2+\\phi{x}{y}={x}^{3}+\\left(\\phi+1\\right){x}^{2}+\\left(6\\phi-19\\right){x}+4\\phi-43',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '2.6947220793073226354123627842800373707',
'lvalue': '0.60255817498007528165769346017631645153',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-79.2-a1',
'class_label': '2.2.5.1-79.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '80.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-8*w+4)',
'conductor_norm': 80,
'ainvs': '0,0;1,0;0,0;-36,0;-140,0',
'equation': '{y}^2={x}^{3}+{x}^{2}-36{x}-140',
'base_change': ['20.a2', '100.a2'],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '0.88634388250184046969156077221643870090',
'lvalue': '0.59457755181456049733689837675530406909',
'sha': 1,
'tamagawa_product': 6,
'label': '2.2.5.1-80.1-a1',
'class_label': '2.2.5.1-80.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '81.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(9)',
'conductor_norm': 81,
'ainvs': '1,0;-1,0;0,1;0,-2;0,1',
'equation': '{y}^2+{x}{y}+\\phi{y}={x}^{3}-{x}^{2}-2\\phi{x}+\\phi',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 6,
'omega': '23.625216251503139776435122373013713579',
'lvalue': '0.58697321690548654029979273929421712342',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-81.1-a1',
'class_label': '2.2.5.1-81.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '89.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w+9)',
'conductor_norm': 89,
'ainvs': '0,1;0,-1;1,0;-31,15;-36,12',
'equation': '{y}^2+\\phi{x}{y}+{y}={x}^{3}-\\phi{x}^{2}+\\left(15\\phi-31\\right){x}+12\\phi-36',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '2.7289225656559698796422383203904987539',
'lvalue': '0.61020563621398816273578938094320315199',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-89.1-a1',
'class_label': '2.2.5.1-89.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '89.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(w-10)',
'conductor_norm': 89,
'ainvs': '1,1;-1,0;1,0;-16,-16;-24,-12',
'equation': '{y}^2+\\left(\\phi+1\\right){x}{y}+{y}={x}^{3}-{x}^{2}+\\left(-16\\phi-16\\right){x}-12\\phi-24',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '2.7289225656559698796422383203904987539',
'lvalue': '0.61020563621398816273578938094320315199',
'sha': 1,
'tamagawa_product': 2,
'label': '2.2.5.1-89.2-a1',
'class_label': '2.2.5.1-89.2-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '95.1',
'class': 'a',
'curve': 1,
'conductor_ideal': '(-2*w+11)',
'conductor_norm': 95,
'ainvs': '0,1;1,1;0,1;-65,32;-238,122',
'equation': '{y}^2+\\phi{x}{y}+\\phi{y}={x}^{3}+\\left(\\phi+1\\right){x}^{2}+\\left(32\\phi-65\\right){x}+122\\phi-238',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
'omega': '1.8889835155647757461823316793024662620',
'lvalue': '0.63358433237690559194859670875150935307',
'sha': 1,
'tamagawa_product': 3,
'label': '2.2.5.1-95.1-a1',
'class_label': '2.2.5.1-95.1-a'},
{'field_label': '2.2.5.1',
'D': 5,
'polynomial': 'y^2-y-1',
'conductor': '95.2',
'class': 'a',
'curve': 1,
'conductor_ideal': '(2*w+9)',
'conductor_norm': 95,
'ainvs': '1,0;1,0;1,1;-497,141;-7218,-2485',
'equation': '{y}^2+{x}{y}+\\left(\\phi+1\\right){y}={x}^{3}+{x}^{2}+\\left(141\\phi-497\\right){x}-2485\\phi-7218',
'base_change': [],
'rank': 0,
'rank_bounds': '[0,0]',
'analytic_rank': 0,
'regulator': '1',
'torsion_order': 2,
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