Elliptic curves by rank › curves from searches

Torsion Z/2xZ/2: record curves

The curves of this collection with torsion subgroup Z/2xZ/2, by rank: for each rank the ten smallest conductors, with generators of the Mordell-Weil group and of the torsion subgroup. Every curve has proven rank, is given by its global minimal model, and appears in the corresponding rank_r file as well. Conductors are exact integers; heights are natural logarithms; the naive height is log max(abs(c4)^3, c6^2).

rank	curves	smallest log N	smallest naive height	smallest log abs(disc)
5	48	22.340206	57.235002	47.668896
6	162	29.049268	70.205769	62.492985

Rank 5

#	a-invariants	conductor N	log N	naive h	log abs(disc)	torsion generators
1	`[1,-1,0,-67864923,-24603903275]`	`5037649974`	22.340205535	65.712693000	58.24479518	`[[8414,-4207],[-8050,4025]]`
2	`[1,-1,0,-6472503,2938419085]`	`6485495346`	22.592834035	58.662823000	50.96641159	`[[-2746,1373],[470,-235]]`
3	`[0,-1,0,-4021369,2911294969]`	`22447563072`	23.834447896	57.235002000	47.66889579	`[[-2299,0],[913,0]]`
4	`[0,0,0,-88808223,120840818722]`	`31345170120`	24.168326029	66.519572000	58.91318612	`[[-10042,0],[1391,0]]`
5	`[0,-1,0,-57610996,145627228420]`	`38142501720`	24.364595028	65.221275000	56.38647285	`[[-8630,0],[2993,0]]`
6	`[0,-1,0,-53026080,143561892672]`	`40027043760`	24.412821157	64.972486000	54.81710686	`[[-8376,0],[3553,0]]`
7	`[0,-1,0,-305210040,1705705257600]`	`48418471920`	24.603147229	70.223135000	61.59510524	`[[-19784,0],[6480,0]]`
8	`[0,1,0,-38270409,86932347831]`	`63013841088`	24.866620239	63.994166000	54.12768963	`[[-7107,0],[2927,0]]`
9	`[1,0,0,-245841163,993128344817]`	`63635893350`	24.876443509	69.574192000	61.52509508	`[[-69617/4,69617/8],[4382,-2191]]`
10	`[1,-1,0,-165073428,798869545600]`	`64707043518`	24.893135897	68.379306000	57.76423989	`[[-14800,7400],[8288,-4144]]`

generators for these ten

1. [1,-1,0,-67864923,-24603903275] N = 5037649974

E = ellinit([1,-1,0,-67864923,-24603903275]);
generators:
  [154826,60756773]
  [-38374/25,34544209/125]
  [-2233,341411]
  [15218,1562981]
  [-7854,-150619]
torsion generators:
  [8414,-4207]
  [-8050,4025]

2. [1,-1,0,-6472503,2938419085] N = 6485495346

E = ellinit([1,-1,0,-6472503,2938419085]);
generators:
  [-2203,81737]
  [8354,725093]
  [2582,57317]
  [-2158,83861]
  [-1942,91421]
torsion generators:
  [-2746,1373]
  [470,-235]

3. [0,-1,0,-4021369,2911294969] N = 22447563072

E = ellinit([0,-1,0,-4021369,2911294969]);
generators:
  [4389,264176]
  [-1463,75240]
  [1463,12540]
  [-1615,72048]
  [2651,104280]
torsion generators:
  [-2299,0]
  [913,0]

4. [0,0,0,-88808223,120840818722] N = 31345170120

E = ellinit([0,0,0,-88808223,120840818722]);
generators:
  [76169,20863062]
  [-11431/4,4741605/8]
  [123553/9,32945374/27]
  [-305731/49,-225455670/343]
  [-3811,635562]
torsion generators:
  [-10042,0]
  [1391,0]

5. [0,-1,0,-57610996,145627228420] N = 38142501720

E = ellinit([0,-1,0,-57610996,145627228420]);
generators:
  [-3792,556370]
  [-8507,141450]
  [11296,967518]
  [159048,63358330]
  [2464,136482]
torsion generators:
  [-8630,0]
  [2993,0]

6. [0,-1,0,-53026080,143561892672] N = 40027043760

E = ellinit([0,-1,0,-53026080,143561892672]);
generators:
  [314,356290]
  [13349,1346950]
  [195474,86364450]
  [114621/4,37653375/8]
  [834896/289,591116680/4913]
torsion generators:
  [-8376,0]
  [3553,0]

7. [0,-1,0,-305210040,1705705257600] N = 48418471920

E = ellinit([0,-1,0,-305210040,1705705257600]);
generators:
  [110,1293110]
  [15190,757770]
  [646665,519830220]
  [56405/4,3627715/8]
  [-4240,1709840]
torsion generators:
  [-19784,0]
  [6480,0]

8. [0,1,0,-38270409,86932347831] N = 63013841088

E = ellinit([0,1,0,-38270409,86932347831]);
generators:
  [-6150,299541]
  [75543,20695560]
  [21081,2940948]
  [-7089,45072]
  [-9965/9,9657928/27]
torsion generators:
  [-7107,0]
  [2927,0]

9. [1,0,0,-245841163,993128344817] N = 63635893350

E = ellinit([1,0,0,-245841163,993128344817]);
generators:
  [902,878249]
  [422678,274399985]
  [52448/49,293119457/343]
  [4394402,9209654249]
  [-17278,-279211]
torsion generators:
  [-69617/4,69617/8]
  [4382,-2191]

10. [1,-1,0,-165073428,798869545600] N = 64707043518

E = ellinit([1,-1,0,-165073428,798869545600]);
generators:
  [5291,268583]
  [6377,70931]
  [1187,776966]
  [2774741/289,1521965104/4913]
  [-52141/4,6913489/8]
torsion generators:
  [-14800,7400]
  [8288,-4144]

Rank 6

#	a-invariants	conductor N	log N	naive h	log abs(disc)	torsion generators
1	`[1,0,0,-303448314,970310480139]`	`4129873336563`	29.049267853	70.205769000	62.49298486	`[[-75361/4,75361/8],[3318,-1659]]`
2	`[0,0,0,-2092695843,6682415701858]`	`4184222602320`	29.062342045	75.998760000	68.51059779	`[[-47266,0],[3209,0]]`
3	`[0,-1,0,-559889409,313921714689]`	`4407664742592`	29.114366128	72.043353000	64.58483992	`[[-23937,0],[561,0]]`
4	`[0,-1,0,-31736142881,874617713691681]`	`10956498893760`	30.024953902	84.155769000	76.52486638	`[[-190591,0],[28271,0]]`
5	`[0,0,0,-6444371892,156806810185376]`	`11336746656960`	30.059070482	79.373022000	70.95034177	`[[-90434,0],[27592,0]]`
6	`[0,-1,0,-3304393108,72733213658212]`	`15460567128600`	30.369313842	77.369159000	65.34379787	`[[-66338,0],[31217,0]]`
7	`[0,-1,0,-6040506761,157443563053065]`	`16028164361280`	30.405368563	79.178864000	70.30060138	`[[-88435,0],[30993,0]]`
8	`[0,-1,0,-3813604608,22432138613712]`	`16595089328400`	30.440127944	77.799125000	70.28121662	`[[-64508,0],[5937,0]]`
9	`[0,-1,0,-4238961433,106151317379737]`	`16884362913600`	30.457409038	78.116355000	64.13369390	`[[-75173,0],[36757,0]]`
10	`[1,0,0,-1034398205,9475169986800]`	`19891394744835`	30.621308329	73.884860000	65.63706581	`[[-144081/4,144081/8],[10180,-5090]]`

generators for these ten

1. [1,0,0,-303448314,970310480139] N = 4129873336563

E = ellinit([1,0,0,-303448314,970310480139]);
generators:
  [495885,348735777]
  [20754,1890147]
  [43719/25,83276193/125]
  [195/4,7819851/8]
  [-2993913/2809,169336442652/148877]
  [19305,1509222]
torsion generators:
  [-75361/4,75361/8]
  [3318,-1659]

2. [0,0,0,-2092695843,6682415701858] N = 4184222602320

E = ellinit([0,0,0,-2092695843,6682415701858]);
generators:
  [-18066,6212300]
  [2009,1576800]
  [-20767,6417576]
  [1535009,1900963800]
  [-41791,4599000]
  [45759,2595550]
torsion generators:
  [-47266,0]
  [3209,0]

3. [0,-1,0,-559889409,313921714689] N = 4407664742592

E = ellinit([0,-1,0,-559889409,313921714689]);
generators:
  [-3165,1433268]
  [-5856,1841679]
  [-22255,1323328]
  [-2161,1230344]
  [44113,7839360]
  [295153689/11881,1706192803584/1295029]
torsion generators:
  [-23937,0]
  [561,0]

4. [0,-1,0,-31736142881,874617713691681] N = 10956498893760

E = ellinit([0,-1,0,-31736142881,874617713691681]);
generators:
  [-5504,32390225]
  [-2604275/36,11413019681/216]
  [179583,31094616]
  [9121,24205600]
  [-33775,43680384]
  [-8017,33593616]
torsion generators:
  [-190591,0]
  [28271,0]

5. [0,0,0,-6444371892,156806810185376] N = 11336746656960

E = ellinit([0,0,0,-6444371892,156806810185376]);
generators:
  [15142,7918200]
  [-15068,15826860]
  [67930,5701104]
  [63142,1279800]
  [576178/9,69504200/27]
  [-9758,14790600]
torsion generators:
  [-90434,0]
  [27592,0]

6. [0,-1,0,-3304393108,72733213658212] N = 15460567128600

E = ellinit([0,-1,0,-3304393108,72733213658212]);
generators:
  [37212,1139050]
  [35692,510150]
  [50728,5970366]
  [456528/49,2242789450/343]
  [456862,306464400]
  [157684/25,902618598/125]
torsion generators:
  [-66338,0]
  [31217,0]

7. [0,-1,0,-6040506761,157443563053065] N = 16028164361280

E = ellinit([0,-1,0,-6040506761,157443563053065]);
generators:
  [198893,82606800]
  [-7624,14249673]
  [68621,8127648]
  [-661763/9,385687000/27]
  [-84007,8487000]
  [266168,131933175]
torsion generators:
  [-88435,0]
  [30993,0]

8. [0,-1,0,-3813604608,22432138613712] N = 16595089328400

E = ellinit([0,-1,0,-3813604608,22432138613712]);
generators:
  [-60888,5375700]
  [596142,457830450]
  [-6108,6745200]
  [-58303,6825500]
  [90977/16,61770775/64]
  [122372,37259200]
torsion generators:
  [-64508,0]
  [5937,0]

9. [0,-1,0,-4238961433,106151317379737] N = 16884362913600

E = ellinit([0,-1,0,-4238961433,106151317379737]);
generators:
  [12936,-7313047]
  [43153,1893216]
  [42567,1684900]
  [-7833,11784500]
  [53767,5802300]
  [40267,865800]
torsion generators:
  [-75173,0]
  [36757,0]

10. [1,0,0,-1034398205,9475169986800] N = 19891394744835

E = ellinit([1,0,0,-1034398205,9475169986800]);
generators:
  [108595/4,9366755/8]
  [-4523,3752092]
  [10135,175540]
  [70624/9,-36743956/27]
  [25927,-304283]
  [7465,1469155]
torsion generators:
  [-144081/4,144081/8]
  [10180,-5090]