Elliptic curves by rank › curves from searches

Rank 8 record curves

The ten smallest curves of rank 8 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 8, is given by its global minimal model, and carries 8 independent generators, listed below each table.

Conductors are given as exact integers. All heights are natural logarithms. The naive height is log max(abs(c4)^3, c6^2); the Faltings height is -(1/2) log abs(Im(conj(w1) w2)) on the minimal model, the normalization used by the Elliptic Curve Rank Leaderboard.

Smallest conductor

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[1,-1,0,-106384,13075804]`	`249649566346838`	33.151079000	46.338040000	1.76394197	35.79013665
2	`[0,0,0,-2220127,1296242146]`	`255180258802520`	33.172991000	55.488616000	2.51442181	44.68591677
3	`[1,-1,0,-222751,40537273]`	`292246301470558`	33.308618000	48.555861000	1.78842300	34.00176524
4	`[1,-1,0,-1336456,594673996]`	`311374862578586`	33.372019000	53.930199000	2.24715946	39.68918333
5	`[0,0,0,-481663,128212738]`	`314214346667560`	33.381097000	50.868603000	2.06064398	38.45627032
6	`[1,-1,0,-71899,5522449]`	`314658846776578`	33.382510000	45.162664000	1.78475360	36.90887067
7	`[1,-1,0,-124294,14418784]`	`315734078239402`	33.385921000	46.804823000	1.89360034	38.04936054
8	`[1,-1,1,-311982,54679485]`	`320064134345894`	33.399543000	49.565702000	2.13546924	41.02416150
9	`[1,-1,0,-201814,34925104]`	`321754587851786`	33.404810000	48.258911000	1.77736928	34.09795740
10	`[0,0,1,-135109,18252072]`	`323954505704623`	33.411624000	47.055114000	1.85577260	37.17282432

generators for these ten

1. [1,-1,0,-106384,13075804] N = 249649566346838

E = ellinit([1,-1,0,-106384,13075804]);
generators:
  [-270,4832]
  [-18,3880]
  [45,2872]
  [122,1318]
  [143,807]
  [158,360]
  [217,308]
  [220,422]

2. [0,0,0,-2220127,1296242146] N = 255180258802520

E = ellinit([0,0,0,-2220127,1296242146]);
generators:
  [-1703,11750]
  [-1653,21200]
  [-453,47000]
  [-53,37600]
  [1397,30350]
  [3697,208850]
  [4667,304280]
  [8497,771950]

3. [1,-1,0,-222751,40537273] N = 292246301470558

E = ellinit([1,-1,0,-222751,40537273]);
generators:
  [-544,981]
  [-511,4809]
  [-413,8078]
  [-408,8165]
  [-364,8743]
  [-318,9055]
  [-281,9134]
  [1404,49237]

4. [1,-1,0,-1336456,594673996] N = 311374862578586

E = ellinit([1,-1,0,-1336456,594673996]);
generators:
  [682,-64]
  [-426,-32750]
  [-149,28190]
  [2451/4,-21287/8]
  [2621,-124160]
  [-980,-30534]
  [5114,354496]
  [1236,28190]

5. [0,0,0,-481663,128212738] N = 314214346667560

E = ellinit([0,0,0,-481663,128212738]);
generators:
  [-801,320]
  [-799,1726]
  [-778,5660]
  [-767,6814]
  [-646,13030]
  [-631,13450]
  [-209,14824]
  [-99,13226]

6. [1,-1,0,-71899,5522449] N = 314658846776578

E = ellinit([1,-1,0,-71899,5522449]);
generators:
  [-209,3479]
  [-73,3258]
  [12,2153]
  [63,1082]
  [216,79]
  [250,1643]
  [267,2170]
  [896,25239]

7. [1,-1,0,-124294,14418784] N = 315734078239402

E = ellinit([1,-1,0,-124294,14418784]);
generators:
  [-399,809]
  [-394,1650]
  [-337,4405]
  [-293,5208]
  [-272,5432]
  [1032,30860]
  [3046,165480]
  [-2690/9,138496/27]

8. [1,-1,1,-311982,54679485] N = 320064134345894

E = ellinit([1,-1,1,-311982,54679485]);
generators:
  [-631,355]
  [-629,1641]
  [-623,2947]
  [-613,4217]
  [-595,5719]
  [-487,9779]
  [5033,352387]
  [-14471/25,837207/125]

9. [1,-1,0,-201814,34925104] N = 321754587851786

E = ellinit([1,-1,0,-201814,34925104]);
generators:
  [-510,2492]
  [-459,5770]
  [-418,7000]
  [-344,8144]
  [-315,8353]
  [-239,8454]
  [777,18233]
  [1078,32160]

10. [0,0,1,-135109,18252072] N = 323954505704623

E = ellinit([0,0,1,-135109,18252072]);
generators:
  [-386,3590]
  [-343,4923]
  [-128,5783]
  [130,1698]
  [173,236]
  [259,795]
  [265,1028]
  [388,4923]

Smallest naive height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,1,1,-16440,1394010]`	`561715239383323`	33.962016000	41.826383000	1.52445428	33.96201615
2	`[0,0,1,-23737,960366]`	`457532830151317`	33.756870000	41.837974000	1.51760795	33.75686976
3	`[0,1,1,-23846,1022562]`	`409086620841461`	33.644948000	41.851760000	1.51152763	33.64494804
4	`[1,0,1,21882,1502972]`	`822049277971034`	34.342821000	41.966628000	1.60525030	35.03596864
5	`[1,-1,1,-10479,1611834]`	`1045057162881491`	34.582848000	42.105659000	1.56466199	34.58284798
6	`[1,0,0,-14470,1656711]`	`993529014001123`	34.532284000	42.165290000	1.56456111	34.53228438
7	`[1,-1,0,-27019,779689]`	`502155474290006`	33.849931000	42.226511000	1.57334538	34.54307808
8	`[1,-1,0,-27589,1109929]`	`409193376769886`	33.645209000	42.289141000	1.56277818	34.33835614
9	`[0,0,1,-28327,538836]`	`1329298918250437`	34.823428000	42.368315000	1.59347513	34.82342807
10	`[0,0,1,-28399,454320]`	`1376680825642789`	34.858452000	42.375931000	1.59576489	34.85845180

generators for these ten

1. [0,1,1,-16440,1394010] N = 561715239383323

E = ellinit([0,1,1,-16440,1394010]);
generators:
  [-159,116]
  [-124,1241]
  [13,1087]
  [44,870]
  [77,768]
  [104,905]
  [107,933]
  [155,1610]

2. [0,0,1,-23737,960366] N = 457532830151317

E = ellinit([0,0,1,-23737,960366]);
generators:
  [-171,138]
  [-44,1385]
  [17,749]
  [33,461]
  [39,306]
  [43,138]
  [130,267]
  [181,1610]

3. [0,1,1,-23846,1022562] N = 409086620841461

E = ellinit([0,1,1,-23846,1022562]);
generators:
  [-93,1563]
  [-62,1505]
  [-50,1446]
  [-5,1068]
  [22,713]
  [47,88]
  [128,289]
  [153,989]

4. [1,0,1,21882,1502972] N = 822049277971034

E = ellinit([1,0,1,21882,1502972]);
generators:
  [-59,115]
  [-57,295]
  [-1,1217]
  [11,1315]
  [22,1401]
  [64,1747]
  [83,1931]
  [197,3571]

5. [1,-1,1,-10479,1611834] N = 1045057162881491

E = ellinit([1,-1,1,-10479,1611834]);
generators:
  [-142,541]
  [-94,1373]
  [-34,1405]
  [61,1063]
  [78,1085]
  [84,1106]
  [93,1151]
  [226,3165]

6. [1,0,0,-14470,1656711] N = 993529014001123

E = ellinit([1,0,0,-14470,1656711]);
generators:
  [70,959]
  [-99,1506]
  [229,3105]
  [-9,1341]
  [53,993]
  [194,-2579]
  [117,-1311]
  [-145,916]

7. [1,-1,0,-27019,779689] N = 502155474290006

E = ellinit([1,-1,0,-27019,779689]);
generators:
  [-176,333]
  [-138,1435]
  [-128,1523]
  [-69,1555]
  [-52,1455]
  [149,138]
  [152,333]
  [180,1223]

8. [1,-1,0,-27589,1109929] N = 409193376769886

E = ellinit([1,-1,0,-27589,1109929]);
generators:
  [-183,163]
  [-177,740]
  [-168,1075]
  [-164,1179]
  [-142,1537]
  [-112,1725]
  [-62,1637]
  [276,3665]

9. [0,0,1,-28327,538836] N = 1329298918250437

E = ellinit([0,0,1,-28327,538836]);
generators:
  [-177,86]
  [-687/4,4645/8]
  [-164,879]
  [-160,987]
  [-158,1034]
  [-129,1430]
  [-32,1188]
  [372,6440]

10. [0,0,1,-28399,454320] N = 1376680825642789

E = ellinit([0,0,1,-28399,454320]);
generators:
  [-14,-922]
  [-87,1505]
  [-75,-1471]
  [-3,734]
  [232,2520]
  [-118,-1471]
  [-168,-696]
  [-172,-501]

Smallest Faltings height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,1,1,-23846,1022562]`	`409086620841461`	33.644948000	41.851760000	1.51152763	33.64494804
2	`[0,0,1,-23737,960366]`	`457532830151317`	33.756870000	41.837974000	1.51760795	33.75686976
3	`[0,1,1,-16440,1394010]`	`561715239383323`	33.962016000	41.826383000	1.52445428	33.96201615
4	`[1,-1,0,-27589,1109929]`	`409193376769886`	33.645209000	42.289141000	1.56277818	34.33835614
5	`[1,0,0,-14470,1656711]`	`993529014001123`	34.532284000	42.165290000	1.56456111	34.53228438
6	`[1,-1,1,-10479,1611834]`	`1045057162881491`	34.582848000	42.105659000	1.56466199	34.58284798
7	`[0,0,1,-22477,1920276]`	`866216414066363`	34.395156000	42.459105000	1.56655982	34.39515589
8	`[1,-1,0,-27019,779689]`	`502155474290006`	33.849931000	42.226511000	1.57334538	34.54307808
9	`[1,0,0,-28772,1126801]`	`973533662904239`	34.511954000	42.415079000	1.57606254	34.51195352
10	`[0,0,1,-28327,538836]`	`1329298918250437`	34.823428000	42.368315000	1.59347513	34.82342807

generators for these ten

1. [0,1,1,-23846,1022562] N = 409086620841461

E = ellinit([0,1,1,-23846,1022562]);
generators:
  [-93,1563]
  [-62,1505]
  [-50,1446]
  [-5,1068]
  [22,713]
  [47,88]
  [128,289]
  [153,989]

2. [0,0,1,-23737,960366] N = 457532830151317

E = ellinit([0,0,1,-23737,960366]);
generators:
  [-171,138]
  [-44,1385]
  [17,749]
  [33,461]
  [39,306]
  [43,138]
  [130,267]
  [181,1610]

3. [0,1,1,-16440,1394010] N = 561715239383323

E = ellinit([0,1,1,-16440,1394010]);
generators:
  [-159,116]
  [-124,1241]
  [13,1087]
  [44,870]
  [77,768]
  [104,905]
  [107,933]
  [155,1610]

4. [1,-1,0,-27589,1109929] N = 409193376769886

E = ellinit([1,-1,0,-27589,1109929]);
generators:
  [-183,163]
  [-177,740]
  [-168,1075]
  [-164,1179]
  [-142,1537]
  [-112,1725]
  [-62,1637]
  [276,3665]

5. [1,0,0,-14470,1656711] N = 993529014001123

E = ellinit([1,0,0,-14470,1656711]);
generators:
  [70,959]
  [-99,1506]
  [229,3105]
  [-9,1341]
  [53,993]
  [194,-2579]
  [117,-1311]
  [-145,916]

6. [1,-1,1,-10479,1611834] N = 1045057162881491

E = ellinit([1,-1,1,-10479,1611834]);
generators:
  [-142,541]
  [-94,1373]
  [-34,1405]
  [61,1063]
  [78,1085]
  [84,1106]
  [93,1151]
  [226,3165]

7. [0,0,1,-22477,1920276] N = 866216414066363

E = ellinit([0,0,1,-22477,1920276]);
generators:
  [-727/4,327/8]
  [-178,530]
  [-176,651]
  [-174,750]
  [-169,944]
  [-163,1119]
  [-559/4,12213/8]
  [-123,1680]

8. [1,-1,0,-27019,779689] N = 502155474290006

E = ellinit([1,-1,0,-27019,779689]);
generators:
  [-176,333]
  [-138,1435]
  [-128,1523]
  [-69,1555]
  [-52,1455]
  [149,138]
  [152,333]
  [180,1223]

9. [1,0,0,-28772,1126801] N = 973533662904239

E = ellinit([1,0,0,-28772,1126801]);
generators:
  [-135,1666]
  [-95,1781]
  [-49,1580]
  [-33,1445]
  [32,473]
  [37,318]
  [437,8270]
  [472,9395]

10. [0,0,1,-28327,538836] N = 1329298918250437

E = ellinit([0,0,1,-28327,538836]);
generators:
  [-177,86]
  [-687/4,4645/8]
  [-164,879]
  [-160,987]
  [-158,1034]
  [-129,1430]
  [-32,1188]
  [372,6440]

Smallest absolute discriminant

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,1,1,-23846,1022562]`	`409086620841461`	33.644948000	41.851760000	1.51152763	33.64494804
2	`[0,0,1,-23737,960366]`	`457532830151317`	33.756870000	41.837974000	1.51760795	33.75686976
3	`[0,1,1,-16440,1394010]`	`561715239383323`	33.962016000	41.826383000	1.52445428	33.96201615
4	`[1,-1,0,-222751,40537273]`	`292246301470558`	33.308618000	48.555861000	1.78842300	34.00176524
5	`[1,-1,0,-201814,34925104]`	`321754587851786`	33.404810000	48.258911000	1.77736928	34.09795740
6	`[1,-1,0,-113014,14706784]`	`349101890324014`	33.486385000	46.526938000	1.70139110	34.17953213
7	`[0,0,1,-216517,38753856]`	`810785363339557`	34.329024000	48.469876000	1.79565952	34.32902448
8	`[1,-1,0,-27589,1109929]`	`409193376769886`	33.645209000	42.289141000	1.56277818	34.33835614
9	`[0,0,1,-22477,1920276]`	`866216414066363`	34.395156000	42.459105000	1.56655982	34.39515589
10	`[0,1,1,-117840,15595860]`	`877917620938123`	34.408574000	46.653209000	1.71607343	34.40857388

generators for these ten

1. [0,1,1,-23846,1022562] N = 409086620841461

E = ellinit([0,1,1,-23846,1022562]);
generators:
  [-93,1563]
  [-62,1505]
  [-50,1446]
  [-5,1068]
  [22,713]
  [47,88]
  [128,289]
  [153,989]

2. [0,0,1,-23737,960366] N = 457532830151317

E = ellinit([0,0,1,-23737,960366]);
generators:
  [-171,138]
  [-44,1385]
  [17,749]
  [33,461]
  [39,306]
  [43,138]
  [130,267]
  [181,1610]

3. [0,1,1,-16440,1394010] N = 561715239383323

E = ellinit([0,1,1,-16440,1394010]);
generators:
  [-159,116]
  [-124,1241]
  [13,1087]
  [44,870]
  [77,768]
  [104,905]
  [107,933]
  [155,1610]

4. [1,-1,0,-222751,40537273] N = 292246301470558

E = ellinit([1,-1,0,-222751,40537273]);
generators:
  [-544,981]
  [-511,4809]
  [-413,8078]
  [-408,8165]
  [-364,8743]
  [-318,9055]
  [-281,9134]
  [1404,49237]

5. [1,-1,0,-201814,34925104] N = 321754587851786

E = ellinit([1,-1,0,-201814,34925104]);
generators:
  [-510,2492]
  [-459,5770]
  [-418,7000]
  [-344,8144]
  [-315,8353]
  [-239,8454]
  [777,18233]
  [1078,32160]

6. [1,-1,0,-113014,14706784] N = 349101890324014

E = ellinit([1,-1,0,-113014,14706784]);
generators:
  [-358,3210]
  [-338,3938]
  [-240,5408]
  [-206,5508]
  [-154,5410]
  [130,1418]
  [291,2384]
  [618,13130]

7. [0,0,1,-216517,38753856] N = 810785363339557

E = ellinit([0,0,1,-216517,38753856]);
generators:
  [275,92]
  [262,-106]
  [277,180]
  [342,2169]
  [309,-1164]
  [1097/4,279/8]
  [246,614]
  [177,-2445]

8. [1,-1,0,-27589,1109929] N = 409193376769886

E = ellinit([1,-1,0,-27589,1109929]);
generators:
  [-183,163]
  [-177,740]
  [-168,1075]
  [-164,1179]
  [-142,1537]
  [-112,1725]
  [-62,1637]
  [276,3665]

9. [0,0,1,-22477,1920276] N = 866216414066363

E = ellinit([0,0,1,-22477,1920276]);
generators:
  [-727/4,327/8]
  [-178,530]
  [-176,651]
  [-174,750]
  [-169,944]
  [-163,1119]
  [-559/4,12213/8]
  [-123,1680]

10. [0,1,1,-117840,15595860] N = 877917620938123

E = ellinit([0,1,1,-117840,15595860]);
generators:
  [-394,1008]
  [-379,2441]
  [-294,4991]
  [-268,5291]
  [-195,5585]
  [-161,5515]
  [-18,4208]
  [288,2372]