Elliptic curves by rank › curves from searches

Rank 5 record curves

The ten smallest curves of rank 5 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 5, is given by its global minimal model, and carries 5 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	`[0,0,1,-79,342]`	`19047851`	16.762465000	25.194228000	0.10809174	16.76246484
2	`[1,0,0,-22,219]`	`20384311`	16.830276000	24.317973000	0.08369007	16.83027609
3	`[0,0,1,-247,1476]`	`22966597`	16.949551000	28.141768000	0.21822767	16.94955141
4	`[1,-1,0,-415,3481]`	`34672310`	17.361452000	29.772761000	0.40394764	19.66403704
5	`[0,0,0,-532,4420]`	`37396136`	17.437078000	30.443534000	0.48866399	20.90281384
6	`[0,1,0,-625,6099]`	`48012824`	17.686979000	31.022106000	0.51968993	21.15271460
7	`[0,0,1,-1117,13026]`	`53149343`	17.788616000	32.668808000	0.69283055	23.48905987
8	`[0,0,1,-2037,35806]`	`53155881`	17.788739000	34.494901000	0.74654879	23.28180075
9	`[1,-1,0,-1234,16624]`	`54285434`	17.809766000	32.968107000	0.64858965	22.39473398
10	`[0,1,1,-100,110]`	`55726757`	17.835971000	25.439097000	0.17920168	17.83597097

generators for these ten

1. [0,0,1,-79,342] N = 19047851

E = ellinit([0,0,1,-79,342]);
generators:
  [0,18]
  [4,9]
  [5,8]
  [10,23]
  [12,33]

2. [1,0,0,-22,219] N = 20384311

E = ellinit([1,0,0,-22,219]);
generators:
  [-7,10]
  [-5,17]
  [-1,16]
  [3,12]
  [5,13]

3. [0,0,1,-247,1476] N = 22966597

E = ellinit([0,0,1,-247,1476]);
generators:
  [-18,9]
  [-16,36]
  [-15,42]
  [-7,53]
  [18,53]

4. [1,-1,0,-415,3481] N = 34672310

E = ellinit([1,-1,0,-415,3481]);
generators:
  [3,46]
  [8,21]
  [12,5]
  [13,6]
  [18,31]

5. [0,0,0,-532,4420] N = 37396136

E = ellinit([0,0,0,-532,4420]);
generators:
  [-26,26]
  [-24,58]
  [-22,74]
  [-16,94]
  [-6,86]

6. [0,1,0,-625,6099] N = 48012824

E = ellinit([0,1,0,-625,6099]);
generators:
  [7,46]
  [11,26]
  [15,18]
  [17,26]
  [19,38]

7. [0,0,1,-1117,13026] N = 53149343

E = ellinit([0,0,1,-1117,13026]);
generators:
  [-9,149]
  [11,45]
  [14,11]
  [24,6]
  [37,149]

8. [0,0,1,-2037,35806] N = 53155881

E = ellinit([0,0,1,-2037,35806]);
generators:
  [-50,112]
  [-47,166]
  [-29,265]
  [-11,238]
  [88,733]

9. [1,-1,0,-1234,16624] N = 54285434

E = ellinit([1,-1,0,-1234,16624]);
generators:
  [-40,48]
  [2,118]
  [16,20]
  [23,-8]
  [30,62]

10. [0,1,1,-100,110] N = 55726757

E = ellinit([0,1,1,-100,110]);
generators:
  [-8,21]
  [-4,21]
  [0,10]
  [1,3]
  [9,4]

Smallest naive height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[1,0,0,-22,219]`	`20384311`	16.830276000	24.317973000	0.08369007	16.83027609
2	`[0,0,1,-79,342]`	`19047851`	16.762465000	25.194228000	0.10809174	16.76246484
3	`[0,1,1,-100,110]`	`55726757`	17.835971000	25.439097000	0.17920168	17.83597097
4	`[0,1,1,-30,390]`	`67445803`	18.026835000	25.507694000	0.18305164	18.02683491
5	`[0,1,1,-112,20]`	`89284469`	18.307338000	25.778015000	0.21543962	18.30733811
6	`[0,-1,1,-82,622]`	`117138251`	18.578865000	26.299739000	0.23802959	18.57886543
7	`[0,0,1,-97,606]`	`100365803`	18.424332000	26.337730000	0.23114074	18.42433210
8	`[0,0,1,-37,636]`	`171637883`	18.960897000	26.434329000	0.26047937	18.96089748
9	`[0,1,1,-142,182]`	`161759189`	18.901619000	26.488118000	0.26761925	18.90161930
10	`[0,0,1,-139,732]`	`59754491`	17.905755000	26.715390000	0.21477647	17.90575491

generators for these ten

1. [1,0,0,-22,219] N = 20384311

E = ellinit([1,0,0,-22,219]);
generators:
  [-7,10]
  [-5,17]
  [-1,16]
  [3,12]
  [5,13]

2. [0,0,1,-79,342] N = 19047851

E = ellinit([0,0,1,-79,342]);
generators:
  [0,18]
  [4,9]
  [5,8]
  [10,23]
  [12,33]

3. [0,1,1,-100,110] N = 55726757

E = ellinit([0,1,1,-100,110]);
generators:
  [-8,21]
  [-4,21]
  [0,10]
  [1,3]
  [9,4]

4. [0,1,1,-30,390] N = 67445803

E = ellinit([0,1,1,-30,390]);
generators:
  [-9,3]
  [-8,13]
  [-7,17]
  [-4,21]
  [6,21]

5. [0,1,1,-112,20] N = 89284469

E = ellinit([0,1,1,-112,20]);
generators:
  [-10,15]
  [-9,19]
  [-4,20]
  [-2,15]
  [-1,11]

6. [0,-1,1,-82,622] N = 117138251

E = ellinit([0,-1,1,-82,622]);
generators:
  [-4,29]
  [2,21]
  [4,18]
  [7,18]
  [11,30]

7. [0,0,1,-97,606] N = 100365803

E = ellinit([0,0,1,-97,606]);
generators:
  [-3,29]
  [-1,26]
  [3,18]
  [6,15]
  [9,21]

8. [0,0,1,-37,636] N = 171637883

E = ellinit([0,0,1,-37,636]);
generators:
  [-8,20]
  [-2,26]
  [3,23]
  [4,23]
  [14,53]

9. [0,1,1,-142,182] N = 161759189

E = ellinit([0,1,1,-142,182]);
generators:
  [-10,26]
  [-4,26]
  [-2,21]
  [1,6]
  [11,8]

10. [0,0,1,-139,732] N = 59754491

E = ellinit([0,0,1,-139,732]);
generators:
  [3,18]
  [4,15]
  [8,11]
  [9,14]
  [13,33]

Smallest Faltings height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[1,0,0,-22,219]`	`20384311`	16.830276000	24.317973000	0.08369007	16.83027609
2	`[0,0,1,-79,342]`	`19047851`	16.762465000	25.194228000	0.10809174	16.76246484
3	`[0,1,1,-100,110]`	`55726757`	17.835971000	25.439097000	0.17920168	17.83597097
4	`[0,1,1,-30,390]`	`67445803`	18.026835000	25.507694000	0.18305164	18.02683491
5	`[0,0,1,-139,732]`	`59754491`	17.905755000	26.715390000	0.21477647	17.90575491
6	`[0,1,1,-112,20]`	`89284469`	18.307338000	25.778015000	0.21543962	18.30733811
7	`[0,0,1,-247,1476]`	`22966597`	16.949551000	28.141768000	0.21822767	16.94955141
8	`[0,0,1,-97,606]`	`100365803`	18.424332000	26.337730000	0.23114074	18.42433210
9	`[0,0,1,-169,930]`	`64921931`	17.988696000	27.194052000	0.23405793	17.98869604
10	`[0,-1,1,-82,622]`	`117138251`	18.578865000	26.299739000	0.23802959	18.57886543

generators for these ten

1. [1,0,0,-22,219] N = 20384311

E = ellinit([1,0,0,-22,219]);
generators:
  [-7,10]
  [-5,17]
  [-1,16]
  [3,12]
  [5,13]

2. [0,0,1,-79,342] N = 19047851

E = ellinit([0,0,1,-79,342]);
generators:
  [0,18]
  [4,9]
  [5,8]
  [10,23]
  [12,33]

3. [0,1,1,-100,110] N = 55726757

E = ellinit([0,1,1,-100,110]);
generators:
  [-8,21]
  [-4,21]
  [0,10]
  [1,3]
  [9,4]

4. [0,1,1,-30,390] N = 67445803

E = ellinit([0,1,1,-30,390]);
generators:
  [-9,3]
  [-8,13]
  [-7,17]
  [-4,21]
  [6,21]

5. [0,0,1,-139,732] N = 59754491

E = ellinit([0,0,1,-139,732]);
generators:
  [3,18]
  [4,15]
  [8,11]
  [9,14]
  [13,33]

6. [0,1,1,-112,20] N = 89284469

E = ellinit([0,1,1,-112,20]);
generators:
  [-10,15]
  [-9,19]
  [-4,20]
  [-2,15]
  [-1,11]

7. [0,0,1,-247,1476] N = 22966597

E = ellinit([0,0,1,-247,1476]);
generators:
  [-18,9]
  [-16,36]
  [-15,42]
  [-7,53]
  [18,53]

8. [0,0,1,-97,606] N = 100365803

E = ellinit([0,0,1,-97,606]);
generators:
  [-3,29]
  [-1,26]
  [3,18]
  [6,15]
  [9,21]

9. [0,0,1,-169,930] N = 64921931

E = ellinit([0,0,1,-169,930]);
generators:
  [-15,9]
  [-14,23]
  [-13,30]
  [20,74]
  [43,270]

10. [0,-1,1,-82,622] N = 117138251

E = ellinit([0,-1,1,-82,622]);
generators:
  [-4,29]
  [2,21]
  [4,18]
  [7,18]
  [11,30]

Smallest absolute discriminant

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,0,1,-79,342]`	`19047851`	16.762465000	25.194228000	0.10809174	16.76246484
2	`[1,0,0,-22,219]`	`20384311`	16.830276000	24.317973000	0.08369007	16.83027609
3	`[0,0,1,-247,1476]`	`22966597`	16.949551000	28.141768000	0.21822767	16.94955141
4	`[0,1,1,-100,110]`	`55726757`	17.835971000	25.439097000	0.17920168	17.83597097
5	`[0,0,1,-139,732]`	`59754491`	17.905755000	26.715390000	0.21477647	17.90575491
6	`[0,0,1,-169,930]`	`64921931`	17.988696000	27.194052000	0.23405793	17.98869604
7	`[0,1,1,-30,390]`	`67445803`	18.026835000	25.507694000	0.18305164	18.02683491
8	`[0,0,1,-301,2052]`	`74129723`	18.121327000	28.776530000	0.29517561	18.12132713
9	`[0,0,1,-457,3786]`	`84602123`	18.253470000	30.001408000	0.34900519	18.25346992
10	`[0,1,1,-112,20]`	`89284469`	18.307338000	25.778015000	0.21543962	18.30733811

generators for these ten

1. [0,0,1,-79,342] N = 19047851

E = ellinit([0,0,1,-79,342]);
generators:
  [0,18]
  [4,9]
  [5,8]
  [10,23]
  [12,33]

2. [1,0,0,-22,219] N = 20384311

E = ellinit([1,0,0,-22,219]);
generators:
  [-7,10]
  [-5,17]
  [-1,16]
  [3,12]
  [5,13]

3. [0,0,1,-247,1476] N = 22966597

E = ellinit([0,0,1,-247,1476]);
generators:
  [-18,9]
  [-16,36]
  [-15,42]
  [-7,53]
  [18,53]

4. [0,1,1,-100,110] N = 55726757

E = ellinit([0,1,1,-100,110]);
generators:
  [-8,21]
  [-4,21]
  [0,10]
  [1,3]
  [9,4]

5. [0,0,1,-139,732] N = 59754491

E = ellinit([0,0,1,-139,732]);
generators:
  [3,18]
  [4,15]
  [8,11]
  [9,14]
  [13,33]

6. [0,0,1,-169,930] N = 64921931

E = ellinit([0,0,1,-169,930]);
generators:
  [-15,9]
  [-14,23]
  [-13,30]
  [20,74]
  [43,270]

7. [0,1,1,-30,390] N = 67445803

E = ellinit([0,1,1,-30,390]);
generators:
  [-9,3]
  [-8,13]
  [-7,17]
  [-4,21]
  [6,21]

8. [0,0,1,-301,2052] N = 74129723

E = ellinit([0,0,1,-301,2052]);
generators:
  [6,21]
  [8,12]
  [9,8]
  [10,6]
  [11,8]

9. [0,0,1,-457,3786] N = 84602123

E = ellinit([0,0,1,-457,3786]);
generators:
  [-24,30]
  [-17,81]
  [-11,86]
  [-4,74]
  [27,105]

10. [0,1,1,-112,20] N = 89284469

E = ellinit([0,1,1,-112,20]);
generators:
  [-10,15]
  [-9,19]
  [-4,20]
  [-2,15]
  [-1,11]