Elliptic curves by rank › curves from searches

Rank 4 record curves

The ten smallest curves of rank 4 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 4, is given by its global minimal model, and carries 4 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,-79,289]`	`234446`	12.364981000	24.729058000	-0.08697564	13.05812774
2	`[0,1,1,-72,210]`	`501029`	13.124419000	24.457458000	-0.09500578	13.12441926
3	`[1,0,1,-35,90]`	`556838`	13.230030000	22.591056000	-0.12142422	13.92317681
4	`[1,0,0,-202,1089]`	`602461`	13.308778000	27.539857000	0.04830278	13.30877821
5	`[1,-1,0,-34,64]`	`614066`	13.327858000	22.209183000	-0.12428690	14.02100487
6	`[0,1,1,-534,4578]`	`709111`	13.471767000	30.457606000	0.28482918	16.03671671
7	`[0,1,1,-22,50]`	`723211`	13.491456000	21.632183000	-0.17313312	13.49145630
8	`[1,-1,1,-762,8505]`	`773926`	13.559232000	31.574741000	0.54067649	21.18385053
9	`[1,0,1,-103,462]`	`787882`	13.577104000	25.831982000	0.14544092	17.10346414
10	`[0,1,1,-32,72]`	`812011`	13.607269000	22.360604000	-0.14515243	13.60726917

generators for these ten

1. [1,-1,0,-79,289] N = 234446

E = ellinit([1,-1,0,-79,289]);
generators:
  [-10,7]
  [-9,19]
  [-8,23]
  [0,17]

2. [0,1,1,-72,210] N = 501029

E = ellinit([0,1,1,-72,210]);
generators:
  [-10,5]
  [-9,14]
  [-7,20]
  [11,29]

3. [1,0,1,-35,90] N = 556838

E = ellinit([1,0,1,-35,90]);
generators:
  [2,4]
  [3,2]
  [5,4]
  [9,18]

4. [1,0,0,-202,1089] N = 602461

E = ellinit([1,0,0,-202,1089]);
generators:
  [-16,25]
  [-8,51]
  [-5,47]
  [23,82]

5. [1,-1,0,-34,64] N = 614066

E = ellinit([1,-1,0,-34,64]);
generators:
  [-6,8]
  [-5,12]
  [-3,13]
  [19,68]

6. [0,1,1,-534,4578] N = 709111

E = ellinit([0,1,1,-534,4578]);
generators:
  [-27,6]
  [-14,97]
  [24,78]
  [38,201]

7. [0,1,1,-22,50] N = 723211

E = ellinit([0,1,1,-22,50]);
generators:
  [-1,8]
  [1,5]
  [3,4]
  [4,6]

8. [1,-1,1,-762,8505] N = 773926

E = ellinit([1,-1,1,-762,8505]);
generators:
  [-31,55]
  [-25,119]
  [-23,127]
  [385,7335]

9. [1,0,1,-103,462] N = 787882

E = ellinit([1,0,1,-103,462]);
generators:
  [-11,22]
  [6,5]
  [9,12]
  [23,90]

10. [0,1,1,-32,72] N = 812011

E = ellinit([0,1,1,-32,72]);
generators:
  [-7,1]
  [-27/4,37/8]
  [-5,11]
  [15,56]

Smallest naive height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,-1,1,28,30]`	`2029691`	14.523394000	21.574288000	-0.10218180	14.52339412
2	`[0,1,1,-22,50]`	`723211`	13.491456000	21.632183000	-0.17313312	13.49145630
3	`[0,0,1,-19,60]`	`1129211`	13.937030000	21.720151000	-0.14681077	13.93702972
4	`[1,0,0,-12,61]`	`1549501`	14.253444000	21.777452000	-0.12940493	14.25344350
5	`[0,1,1,-30,0]`	`1740197`	14.369509000	21.850345000	-0.11248234	14.36950888
6	`[1,-1,1,-31,6]`	`1848689`	14.429987000	21.885169000	-0.10802355	14.42998730
7	`[0,1,1,16,72]`	`2184811`	14.597040000	21.932254000	-0.10139784	14.59703988
8	`[1,0,0,-25,66]`	`1000033`	13.815544000	21.964644000	-0.14587736	13.81554356
9	`[0,1,1,-32,30]`	`1437509`	14.178422000	22.041899000	-0.11937365	14.17842231
10	`[0,-1,1,-2,72]`	`2208251`	14.607711000	22.062799000	-0.10370793	14.60771136

generators for these ten

1. [0,-1,1,28,30] N = 2029691

E = ellinit([0,-1,1,28,30]);
generators:
  [-1,0]
  [0,5]
  [2,9]
  [20,90]

2. [0,1,1,-22,50] N = 723211

E = ellinit([0,1,1,-22,50]);
generators:
  [-1,8]
  [1,5]
  [3,4]
  [4,6]

3. [0,0,1,-19,60] N = 1129211

E = ellinit([0,0,1,-19,60]);
generators:
  [-3,9]
  [1,6]
  [2,5]
  [3,5]

4. [1,0,0,-12,61] N = 1549501

E = ellinit([1,0,0,-12,61]);
generators:
  [-5,4]
  [-4,9]
  [-3,10]
  [4,7]

5. [0,1,1,-30,0] N = 1740197

E = ellinit([0,1,1,-30,0]);
generators:
  [-6,0]
  [-4,8]
  [-2,7]
  [-1,5]

6. [1,-1,1,-31,6] N = 1848689

E = ellinit([1,-1,1,-31,6]);
generators:
  [-3,9]
  [-2,8]
  [0,2]
  [12,29]

7. [0,1,1,16,72] N = 2184811

E = ellinit([0,1,1,16,72]);
generators:
  [-3,2]
  [0,8]
  [1,9]
  [3,12]

8. [1,0,0,-25,66] N = 1000033

E = ellinit([1,0,0,-25,66]);
generators:
  [1,6]
  [2,4]
  [3,3]
  [5,6]

9. [0,1,1,-32,30] N = 1437509

E = ellinit([0,1,1,-32,30]);
generators:
  [-5,9]
  [-2,9]
  [0,5]
  [1,0]

10. [0,-1,1,-2,72] N = 2208251

E = ellinit([0,-1,1,-2,72]);
generators:
  [-4,0]
  [-3,6]
  [-1,8]
  [0,8]

Smallest Faltings height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,1,1,-22,50]`	`723211`	13.491456000	21.632183000	-0.17313312	13.49145630
2	`[0,0,1,-19,60]`	`1129211`	13.937030000	21.720151000	-0.14681077	13.93702972
3	`[1,0,0,-25,66]`	`1000033`	13.815544000	21.964644000	-0.14587736	13.81554356
4	`[0,1,1,-32,72]`	`812011`	13.607269000	22.360604000	-0.14515243	13.60726917
5	`[1,0,0,-12,61]`	`1549501`	14.253444000	21.777452000	-0.12940493	14.25344350
6	`[1,-1,0,-34,64]`	`614066`	13.327858000	22.209183000	-0.12428690	14.02100487
7	`[1,0,1,-35,90]`	`556838`	13.230030000	22.591056000	-0.12142422	13.92317681
8	`[0,1,1,-32,30]`	`1437509`	14.178422000	22.041899000	-0.11937365	14.17842231
9	`[0,1,1,-30,0]`	`1740197`	14.369509000	21.850345000	-0.11248234	14.36950888
10	`[1,0,0,-49,120]`	`887657`	13.696341000	23.290339000	-0.11015043	13.69634069

generators for these ten

1. [0,1,1,-22,50] N = 723211

E = ellinit([0,1,1,-22,50]);
generators:
  [-1,8]
  [1,5]
  [3,4]
  [4,6]

2. [0,0,1,-19,60] N = 1129211

E = ellinit([0,0,1,-19,60]);
generators:
  [-3,9]
  [1,6]
  [2,5]
  [3,5]

3. [1,0,0,-25,66] N = 1000033

E = ellinit([1,0,0,-25,66]);
generators:
  [1,6]
  [2,4]
  [3,3]
  [5,6]

4. [0,1,1,-32,72] N = 812011

E = ellinit([0,1,1,-32,72]);
generators:
  [-7,1]
  [-27/4,37/8]
  [-5,11]
  [15,56]

5. [1,0,0,-12,61] N = 1549501

E = ellinit([1,0,0,-12,61]);
generators:
  [-5,4]
  [-4,9]
  [-3,10]
  [4,7]

6. [1,-1,0,-34,64] N = 614066

E = ellinit([1,-1,0,-34,64]);
generators:
  [-6,8]
  [-5,12]
  [-3,13]
  [19,68]

7. [1,0,1,-35,90] N = 556838

E = ellinit([1,0,1,-35,90]);
generators:
  [2,4]
  [3,2]
  [5,4]
  [9,18]

8. [0,1,1,-32,30] N = 1437509

E = ellinit([0,1,1,-32,30]);
generators:
  [-5,9]
  [-2,9]
  [0,5]
  [1,0]

9. [0,1,1,-30,0] N = 1740197

E = ellinit([0,1,1,-30,0]);
generators:
  [-6,0]
  [-4,8]
  [-2,7]
  [-1,5]

10. [1,0,0,-49,120] N = 887657

E = ellinit([1,0,0,-49,120]);
generators:
  [1,8]
  [3,0]
  [5,0]
  [12,30]

Smallest absolute discriminant

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[1,-1,0,-79,289]`	`234446`	12.364981000	24.729058000	-0.08697564	13.05812774
2	`[0,1,1,-72,210]`	`501029`	13.124419000	24.457458000	-0.09500578	13.12441926
3	`[1,0,0,-202,1089]`	`602461`	13.308778000	27.539857000	0.04830278	13.30877821
4	`[0,1,1,-22,50]`	`723211`	13.491456000	21.632183000	-0.17313312	13.49145630
5	`[0,1,1,-32,72]`	`812011`	13.607269000	22.360604000	-0.14515243	13.60726917
6	`[1,0,0,-49,120]`	`887657`	13.696341000	23.290339000	-0.11015043	13.69634069
7	`[0,0,1,-67,216]`	`953243`	13.767625000	24.276016000	-0.07305798	13.76762513
8	`[1,0,0,-25,66]`	`1000033`	13.815544000	21.964644000	-0.14587736	13.81554356
9	`[0,0,1,-91,330]`	`1112437`	13.922064000	25.146182000	-0.03281717	13.92206366
10	`[1,0,1,-35,90]`	`556838`	13.230030000	22.591056000	-0.12142422	13.92317681

generators for these ten

1. [1,-1,0,-79,289] N = 234446

E = ellinit([1,-1,0,-79,289]);
generators:
  [-10,7]
  [-9,19]
  [-8,23]
  [0,17]

2. [0,1,1,-72,210] N = 501029

E = ellinit([0,1,1,-72,210]);
generators:
  [-10,5]
  [-9,14]
  [-7,20]
  [11,29]

3. [1,0,0,-202,1089] N = 602461

E = ellinit([1,0,0,-202,1089]);
generators:
  [-16,25]
  [-8,51]
  [-5,47]
  [23,82]

4. [0,1,1,-22,50] N = 723211

E = ellinit([0,1,1,-22,50]);
generators:
  [-1,8]
  [1,5]
  [3,4]
  [4,6]

5. [0,1,1,-32,72] N = 812011

E = ellinit([0,1,1,-32,72]);
generators:
  [-7,1]
  [-27/4,37/8]
  [-5,11]
  [15,56]

6. [1,0,0,-49,120] N = 887657

E = ellinit([1,0,0,-49,120]);
generators:
  [1,8]
  [3,0]
  [5,0]
  [12,30]

7. [0,0,1,-67,216] N = 953243

E = ellinit([0,0,1,-67,216]);
generators:
  [-9,9]
  [-8,15]
  [-7,18]
  [-4,20]

8. [1,0,0,-25,66] N = 1000033

E = ellinit([1,0,0,-25,66]);
generators:
  [1,6]
  [2,4]
  [3,3]
  [5,6]

9. [0,0,1,-91,330] N = 1112437

E = ellinit([0,0,1,-91,330]);
generators:
  [-11,0]
  [-9,20]
  [1,15]
  [27,132]

10. [1,0,1,-35,90] N = 556838

E = ellinit([1,0,1,-35,90]);
generators:
  [2,4]
  [3,2]
  [5,4]
  [9,18]