Elliptic curves by rank › curves from parametrized families

Family curves, torsion subgroup Z/6: record curves

The curves of the family collection with torsion subgroup Z/6, 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 has rank larger than the generic rank of the family it comes from. 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)
4	14	22.973416	81.190454	69.766723
5	1	30.340903	102.031571	94.790153

Rank 4

#	a-invariants	conductor N	log N	naive h	log abs(disc)	torsion generators
1	`[1,0,1,-97039305368,11609649895462806]`	`9489161630`	22.973416103	87.508749000	74.62019017	`[[150765,20084627]]`
2	`[1,0,0,-315075824080,119503941430790400]`	`11657817490`	23.179242821	92.167397000	84.32041730	`[[780080,589879200]]`
3	`[1,0,0,86716282125,47141165433705057]`	`32634172230`	24.208625804	90.306991000	82.89482378	`[[820974,819143193]]`
4	`[1,0,1,-11735992013,494047350675156]`	`52490555390`	24.683899093	81.190454000	69.76672292	`[[79785,8058002]]`
5	`[1,-1,0,-895277615340,345550319537517756]`	`139401605970`	25.660624856	94.290974000	84.62604220	`[[833550,422012334]]`
6	`[1,0,1,-22390683783,1171200770867318]`	`180119766970`	25.916887839	83.109335000	73.91260265	`[[38069,19319570]]`
7	`[1,0,0,-363963846271,88927256172203801]`	`247236947790`	26.233613017	91.576336000	81.78614382	`[[522686,203421437]]`
8	`[1,-1,1,-41078225312,4272996391666499]`	`338866139430`	26.548870997	85.505324000	77.22408619	`[[234897,86972281]]`
9	`[1,0,1,11973367542,20806773168598268]`	`340241904730`	26.552922686	88.671255000	81.21712265	`[[565229,455958625]]`
10	`[1,0,1,163384132347,22652275615552756]`	`422056123770`	26.768404137	89.071713000	82.20151883	`[[914570,967561347]]`

generators for these ten

1. [1,0,1,-97039305368,11609649895462806] N = 9489161630

E = ellinit([1,0,1,-97039305368,11609649895462806]);
generators:
  [-301115,116458307]
  [769915/4,62536831/8]
  [290654445/1681,29400656187/68921]
  [961606465/5041,2230718954977/357911]
torsion generators:
  [150765,20084627]

2. [1,0,0,-315075824080,119503941430790400] N = 11657817490

E = ellinit([1,0,0,-315075824080,119503941430790400]);
generators:
  [542260,328515020]
  [4626560,9881692720]
  [-20398710/49,144995633460/343]
  [28520864/49,125134286416/343]
torsion generators:
  [780080,589879200]

3. [1,0,0,86716282125,47141165433705057] N = 32634172230

E = ellinit([1,0,0,86716282125,47141165433705057]);
generators:
  [15272334,59687880153]
  [-2398304634/26569,850810162783251/4330747]
  [-31449858/361,1353545810067/6859]
  [114248574/361,2235021323667/6859]
torsion generators:
  [820974,819143193]

4. [1,0,1,-11735992013,494047350675156] N = 52490555390

E = ellinit([1,0,1,-11735992013,494047350675156]);
generators:
  [-18069325/484,316097641201/10648]
  [21816240,101886822587]
  [-1223,22548308]
  [-52583805/676,538487212567/17576]
torsion generators:
  [79785,8058002]

5. [1,-1,0,-895277615340,345550319537517756] N = 139401605970

E = ellinit([1,-1,0,-895277615340,345550319537517756]);
generators:
  [-4295613/4,2097981933/8]
  [1815090,2167143714]
  [4899930,10656006426]
  [-933879477/961,16273323388341/29791]
torsion generators:
  [833550,422012334]

6. [1,0,1,-22390683783,1171200770867318] N = 180119766970

E = ellinit([1,0,1,-22390683783,1171200770867318]);
generators:
  [-108131,48303720]
  [520529,361062070]
  [-180044/9,1084015930/27]
  [941709/16,482341915/64]
torsion generators:
  [38069,19319570]

7. [1,0,0,-363963846271,88927256172203801] N = 247236947790

E = ellinit([1,0,0,-363963846271,88927256172203801]);
generators:
  [-357914,416526637]
  [-307594,414614477]
  [290636,87582077]
  [336206,67384277]
torsion generators:
  [522686,203421437]

8. [1,-1,1,-41078225312,4272996391666499] N = 338866139430

E = ellinit([1,-1,1,-41078225312,4272996391666499]);
generators:
  [-163683,81392161]
  [326877,160372321]
  [899197,832848321]
  [2254369,3370633753]
torsion generators:
  [234897,86972281]

9. [1,0,1,11973367542,20806773168598268] N = 340241904730

E = ellinit([1,0,1,11973367542,20806773168598268]);
generators:
  [-97349/4,1145549835/8]
  [-2259659/9,1202840795/27]
  [41224981/9,264734254075/27]
  [288869329/576,5402883276725/13824]
torsion generators:
  [565229,455958625]

10. [1,0,1,163384132347,22652275615552756] N = 422056123770

E = ellinit([1,0,1,163384132347,22652275615552756]);
generators:
  [9024425,27133072752]
  [-12504822848005/206037316,330910412413105020723/2957459633864]
  [891572512469/94864,842502967585576023/29218112]
  [26532549778739/2002225,136713975868668180432/2833148375]
torsion generators:
  [914570,967561347]

Rank 5

#	a-invariants	conductor N	log N	naive h	log abs(disc)	torsion generators
1	`[1,0,0,7610029720215,16571446920036651097]`	`15027506235170`	30.340903387	102.031571000	94.79015254	`[[6555354,18655739843]]`

generators for these ten

1. [1,0,0,7610029720215,16571446920036651097] N = 15027506235170

E = ellinit([1,0,0,7610029720215,16571446920036651097]);
generators:
  [-660998,3354794755]
  [7539194,22412040963]
  [201279994,2855793744643]
  [9609418/9,137431182161/27]
  [1043822026/9,33732198514001/27]
torsion generators:
  [6555354,18655739843]