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Elliptic curves by rank › curves from parametrized families

Family curves, torsion subgroup Z/4: record curves

The curves of the family collection with torsion subgroup Z/4, 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)
5 171,990 24.649019 81.521171 73.300221
6 115,242 32.690057 92.114991 84.659857
7 50,691 40.746891 111.009303 102.362683
8 11,555 48.168475 139.193599 125.842080
9 688 61.547535 165.613694 154.177092
10 18 78.506308 212.450966 198.295734

Rank 5

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [0,1,0,-1304100895809,573209971740588735] 50691232704 24.649018808 95.303208000 75.38242783 [[605679,74400048]]
2 [0,-1,0,-724633759408,237421632436621312] 319688007600 26.490611381 93.540400000 75.60782820 [[426312,77331400]]
3 [1,1,1,-5209497434719,4576582200875682572] 874968923829 27.497454207 99.458117000 78.75574256 [[1408174,181092846]]
4 [0,-1,0,-2331079809,582886662468993] 1076323152576 27.704571860 81.521171000 74.06091205 [[60513,25756752]]
5 [1,1,1,-15832236526619,24246684049136974697] 1541677902414 28.063892486 102.792811000 85.25208423 [[1962099,856938454]]
6 [1,1,1,-61141279938,5805411313781406] 1721096402025 28.173982647 86.122962000 73.30022098 [[78450,38583287]]
7 [0,0,0,-99187806397836,444377610382002771440] 2966511250368 28.718407715 108.609558000 99.83771982 [[5959198,8057364048]]
8 [0,0,0,-5000000182131,21129879942026543410] 5200575046128 29.279790321 102.517585000 95.02050303 [[2012873,4384174608]]
9 [0,1,0,-4496481981633,3668405303102824863] 7708213780800 29.673307601 99.016552000 84.46554230 [[856923,666723600]]
10 [0,1,0,610943673375,19381525720812834975] 8869566229440 29.813647008 102.344850000 94.89021950 [[1977975,5322458160]]
generators for these ten

1. [0,1,0,-1304100895809,573209971740588735] N = 50691232704

E = ellinit([0,1,0,-1304100895809,573209971740588735]);
generators:
  [569327,123645648]
  [634434,34759683]
  [2645301/4,20975601/8]
  [31829241/64,111762495789/512]
  [370004223/529,693383647344/12167]
torsion generators:
  [605679,74400048]

2. [0,-1,0,-724633759408,237421632436621312] N = 319688007600

E = ellinit([0,-1,0,-724633759408,237421632436621312]);
generators:
  [-836638,507997350]
  [3146512,5395322400]
  [25145898/49,9086449450/343]
  [130341736/625,4825358723016/15625]
  [3100629522/5929,17620807815458/456533]
torsion generators:
  [426312,77331400]

3. [1,1,1,-5209497434719,4576582200875682572] N = 874968923829

E = ellinit([1,1,1,-5209497434719,4576582200875682572]);
generators:
  [981800,638482116]
  [7331315/4,8702829309/8]
  [707043275/484,3075462377259/10648]
  [3821541695/3364,68756578366767/195112]
  [48358960796/36481,103068272276085/6967871]
torsion generators:
  [1408174,181092846]

4. [0,-1,0,-2331079809,582886662468993] N = 1076323152576

E = ellinit([0,-1,0,-2331079809,582886662468993]);
generators:
  [-8577,24540768]
  [-407692439/6889,12957888144720/571787]
  [-3275967/289,121128518064/4913]
  [1513897/9,1899665776/27]
  [235301553/49,3609244772496/343]
torsion generators:
  [60513,25756752]

5. [1,1,1,-15832236526619,24246684049136974697] N = 1541677902414

E = ellinit([1,1,1,-15832236526619,24246684049136974697]);
generators:
  [1645021,1628259278]
  [2232395,166872766]
  [895628249,26802798443104]
  [601555081/225,3418679631184/3375]
  [1725722279/361,52132195939946/6859]
torsion generators:
  [1962099,856938454]

6. [1,1,1,-61141279938,5805411313781406] N = 1721096402025

E = ellinit([1,1,1,-61141279938,5805411313781406]);
generators:
  [123550,11658587]
  [-285056,-8308987]
  [9564625/64,-1278722981/512]
  [25141900/169,-3345324286/2197]
  [594691/4,9517381/8]
torsion generators:
  [78450,38583287]

7. [0,0,0,-99187806397836,444377610382002771440] N = 2966511250368

E = ellinit([0,0,0,-99187806397836,444377610382002771440]);
generators:
  [-8920514,24886318320]
  [4243390,9994651920]
  [14643262,46171720944]
  [17482888,63670691988]
  [19527312206/3721,1876956143200400/226981]
torsion generators:
  [5959198,8057364048]

8. [0,0,0,-5000000182131,21129879942026543410] N = 5200575046128

E = ellinit([0,0,0,-5000000182131,21129879942026543410]);
generators:
  [504729,4328373280]
  [76743122,672024219174]
  [124172537,1383471768096]
  [11221979033/2209,1169264947812192/103823]
  [9889088048777/638401,30861626227276602384/510082399]
torsion generators:
  [2012873,4384174608]

9. [0,1,0,-4496481981633,3668405303102824863] N = 7708213780800

E = ellinit([0,1,0,-4496481981633,3668405303102824863]);
generators:
  [-284727,2219367600]
  [1296237,133706508]
  [-9170283/4,11106568575/8]
  [5094501/4,695059353/8]
  [392689653/361,1739981915100/6859]
torsion generators:
  [856923,666723600]

10. [0,1,0,610943673375,19381525720812834975] N = 8869566229440

E = ellinit([0,1,0,610943673375,19381525720812834975]);
generators:
  [578295,4464104400]
  [2816055,6590430000]
  [18214263,77931138096]
  [2617848783/841,174929268031056/24389]
  [41686916391/625,8512236076562064/15625]
torsion generators:
  [1977975,5322458160]

Rank 6

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,1,0,-13831444993436,19796596662682167156] 157438656284322 32.690057014 102.387500000 86.71817046 [[2680984,1407476442]]
2 [0,0,0,-11417299959756,14781466634060057456] 194542926481728 32.901673956 101.812056000 89.65313849 [[3188014,3283918704]]
3 [0,0,0,1745421357300,128967401282614000] 198468505684800 32.921651542 96.177654000 88.74382823 [[1253630,2070572400]]
4 [1,-1,1,-8506987562381,9545861372354942901] 229617923822118 33.067437843 100.929330000 86.46793268 [[2258447,1360000104]]
5 [0,-1,0,19385373544735,1244196470298701616225] 524976437012160 33.894374496 110.668699000 103.21467627 [[7976905,43662458000]]
6 [1,1,1,-2154476322728455,38491132722091660492052] 688657878546495 34.165765716 117.532575000 89.03428794 [[26540002,2300639811]]
7 [1,1,0,-51628280214000,142779970582531222500] 794246210269050 34.308414618 106.338875000 89.10114334 [[4836000,2488498650]]
8 [0,-1,0,-33560906609,116413230669128289] 841164882657312 34.365808812 92.114991000 84.65985726 [[355045,386333228]]
9 [1,0,1,-1945708911820813,32602171221875725347656] 924252927287910 34.460006881 117.226811000 106.12206483 [[8459875,129406689062]]
10 [1,-1,1,-28076865892061,57262483766694761156] 981381897019899 34.519982793 104.511504000 84.69218757 [[3321726,804745810]]
generators for these ten

1. [1,1,0,-13831444993436,19796596662682167156] N = 157438656284322

E = ellinit([1,1,0,-13831444993436,19796596662682167156]);
generators:
  [24682172839/11236,135665313690287/1191016]
  [332990925651/46225,170366757305903556/9938375]
  [17495910109177/3775249,54497163131474737087/7335308807]
  [62866385506261/28976689,3483770290682607759/155981516887]
  [343160639128774/95746225,3776501019322482219348/936876811625]
  [11606123351932840/5569786161,62592102278530298182766/415678710981591]
torsion generators:
  [2680984,1407476442]

2. [0,0,0,-11417299959756,14781466634060057456] N = 194542926481728

E = ellinit([0,0,0,-11417299959756,14781466634060057456]);
generators:
  [645838,2770762320]
  [3069115886/1681,10267480088624/68921]
  [16004444734/19321,6519132594056016/2685619]
  [347085081022/316969,338232146933316240/178453547]
  [2892327629038/3996001,20977847411002788240/7988005999]
  [182039598413587909/291257740489,441515209433797663971483303/157186851160324987]
torsion generators:
  [3188014,3283918704]

3. [0,0,0,1745421357300,128967401282614000] N = 198468505684800

E = ellinit([0,0,0,1745421357300,128967401282614000]);
generators:
  [784830,1407923600]
  [38000990,234398689200]
  [77705534,685079140752]
  [4885316/49,189110155536/343]
  [12698446/9,62745068656/27]
  [119020145/1024,18920761180425/32768]
torsion generators:
  [1253630,2070572400]

4. [1,-1,1,-8506987562381,9545861372354942901] N = 229617923822118

E = ellinit([1,-1,1,-8506987562381,9545861372354942901]);
generators:
  [1714865,22244802]
  [1756679,150070200]
  [2007563,746421468]
  [-407292577/121,276167385768/1331]
  [30285611/25,126051241044/125]
  [304875869/361,11805325949130/6859]
torsion generators:
  [2258447,1360000104]

5. [0,-1,0,19385373544735,1244196470298701616225] N = 524976437012160

E = ellinit([0,-1,0,19385373544735,1244196470298701616225]);
generators:
  [-1712471055/169,5953372446000/2197]
  [-485367320/81,22025912978125/729]
  [13768945/9,965012850800/27]
  [37612645/4,380139127475/8]
  [158158745/529,430171674870000/12167]
  [4834553295/289,388403122422000/4913]
torsion generators:
  [7976905,43662458000]

6. [1,1,1,-2154476322728455,38491132722091660492052] N = 688657878546495

E = ellinit([1,1,1,-2154476322728455,38491132722091660492052]);
generators:
  [93050483/4,247897134813/8]
  [25762577,9215390511]
  [1312266673/49,49303723573/343]
  [170540258858/2809,53976600921098442/148877]
  [627283952608/23409,7718075611667/3581577]
  [931203571148/37249,114571933306777827/7189057]
torsion generators:
  [26540002,2300639811]

7. [1,1,0,-51628280214000,142779970582531222500] N = 794246210269050

E = ellinit([1,1,0,-51628280214000,142779970582531222500]);
generators:
  [16316475/4,1866967725/8]
  [4120304,74045826]
  [4463969,1123178701]
  [4467750,1136873850]
  [755892375/1444,590713994342325/54872]
  [7374897075/1849,44097736625775/79507]
torsion generators:
  [4836000,2488498650]

8. [0,-1,0,-33560906609,116413230669128289] N = 841164882657312

E = ellinit([0,-1,0,-33560906609,116413230669128289]);
generators:
  [-395573,260365880]
  [-78064,344321655]
  [-30172063433/68121,3745856362977100/17779581]
  [25925/4,2727002589/8]
  [305926271/625,7283024081556/15625]
  [12654682343/162409,22125020012637420/65450827]
torsion generators:
  [355045,386333228]

9. [1,0,1,-1945708911820813,32602171221875725347656] N = 924252927287910

E = ellinit([1,0,1,-1945708911820813,32602171221875725347656]);
generators:
  [23041387,1760133014]
  [71193700,504868631687]
  [5192570875/49,345790970589266/343]
  [118665129475/2809,23908425777527374/148877]
  [7548752897125/44944,20085186954346500061/9528128]
  [905875757572750/39400729,1129868061182943853346/247318375933]
torsion generators:
  [8459875,129406689062]

10. [1,-1,1,-28076865892061,57262483766694761156] N = 981381897019899

E = ellinit([1,-1,1,-28076865892061,57262483766694761156]);
generators:
  [321081,6948266383]
  [3855247/4,44607882359/8]
  [1580262,4102860943]
  [267941643/4,4372323696809/8]
  [-44994638/9,230241098261/27]
  [49265061/16,3683854927/64]
torsion generators:
  [3321726,804745810]

Rank 7

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,-1,1,-14967172566708386,704617133741140511012001] 496764009210534318 40.746891478 123.347510000 108.25403113 [[52011521,258533852895]]
2 [1,1,1,-217075998668420,1475197160196665902820] 526770622069324545 40.805541596 111.009303000 102.36268286 [[8868320,15729779659]]
3 [0,0,0,-5679832676302668,164482318080161433485008] 1012399021707226560 41.458854457 120.440698000 107.29202645 [[25343234,191870866440]]
4 [1,-1,1,-109108090361789456,16610051049747481430026755] 1354000263461127126 41.749595043 129.667246000 121.01695320 [[198783965,1666055898165]]
5 [0,-1,0,-750826076561680,11332358425729416579700] 1578385609386038880 41.902934232 115.087044000 106.96233576 [[17228052,59249242838]]
6 [1,1,1,-210050813820678350,229959419870628137482189067] 1906431864891437910 42.091765035 134.923036000 127.44200904 [[444846737,14984740293431]]
7 [1,1,1,-30152812499704430,1992633115588943693459627] 1936900914619005510 42.107620903 125.448767000 114.19396568 [[35540177,982778124311]]
8 [1,1,0,-5157621363938497,203250515623581340378681] 2011020983594216490 42.145174219 120.860598000 112.72856819 [[45110050,249747640609]]
9 [1,-1,1,-89370379241164742,10520290211083247769005909] 2698227642570635610 42.439126803 128.753842000 118.18730538 [[173477817,486995646091]]
10 [1,1,1,700175330673221,16719390490661090685920] 4272863122202808237 42.898815797 115.864858000 108.57727634 [[22569826,209796315511]]
generators for these ten

1. [1,-1,1,-14967172566708386,704617133741140511012001] N = 496764009210534318

E = ellinit([1,-1,1,-14967172566708386,704617133741140511012001]);
generators:
  [71818061,11341529943]
  [-64083220131551935/530887681,10611501328654306464397407/12232183057921]
  [45400146377/841,5664539662812891/24389]
  [379615640129/16129,1238162820784061985/2048383]
  [5526686987177/26569,11118530630975032581/4330747]
  [247315635024385/3448449,57371398840704875231/6403769793]
  [143758823299434321315593/86234197620169,54363450713285060259474982088167899/800791196605725314053]
torsion generators:
  [52011521,258533852895]

2. [1,1,1,-217075998668420,1475197160196665902820] N = 526770622069324545

E = ellinit([1,1,1,-217075998668420,1475197160196665902820]);
generators:
  [-2992144239112/2193361,136618752273576291323/3248367641]
  [-906085517/81,36481078215422/729]
  [21663521376110/32041,100805243520592187555/5735339]
  [6382410131798320/20820969,509299870343990692776265/95006081547]
  [10978932388407232/546297129,923244303477017871921671/12768602796117]
  [25279721799429232/3341649249,3149518437906086342220524/193170718136943]
  [13456551732858453332/520898523289,43037005371582297864541349876/375949653908939837]
torsion generators:
  [8868320,15729779659]

3. [0,0,0,-5679832676302668,164482318080161433485008] N = 1012399021707226560

E = ellinit([0,0,0,-5679832676302668,164482318080161433485008]);
generators:
  [-7354596,453715089080]
  [40983573,23252371681]
  [45200709,9980366935]
  [45272429,11512700595]
  [-3691258616718/51529,5280049522681380392/11697083]
  [1988518002/49,9808794920408/343]
  [82620195244/2025,2345279850848528/91125]
torsion generators:
  [25343234,191870866440]

4. [1,-1,1,-109108090361789456,16610051049747481430026755] N = 1354000263461127126

E = ellinit([1,-1,1,-109108090361789456,16610051049747481430026755]);
generators:
  [-379900897,1797752901801]
  [-359499837,3061647531345]
  [-298303117,4755452112795]
  [70799951,3039717512649]
  [-651036216541/1681,60694106738450307/68921]
  [-99115571749/625,85475371027991049/15625]
  [441484866923/1681,169794081523977945/68921]
torsion generators:
  [198783965,1666055898165]

5. [0,-1,0,-750826076561680,11332358425729416579700] N = 1578385609386038880

E = ellinit([0,-1,0,-750826076561680,11332358425729416579700]);
generators:
  [119492417,1275844565658]
  [-2765676724470495/287675521,648449096088957876959870/4879264511681]
  [-166805600711/7744,90179158020909645/681472]
  [-19034700983/576,103232596200283/13824]
  [-330034023/16,8596833630347/64]
  [19180783677/961,1952982748805552/29791]
  [25811558356/225,4042842304672054/3375]
torsion generators:
  [17228052,59249242838]

6. [1,1,1,-210050813820678350,229959419870628137482189067] N = 1906431864891437910

E = ellinit([1,1,1,-210050813820678350,229959419870628137482189067]);
generators:
  [-706779663,5035839823831]
  [-663593673,8782699502221]
  [122324777,14355961598551]
  [-4401233669378111/26656569,2219789865056601934420853/137627865747]
  [-38177632703/81,10917085684656799/729]
  [-6476179607/9,78813723800477/27]
  [152929236103/324,89586046963127087/5832]
torsion generators:
  [444846737,14984740293431]

7. [1,1,1,-30152812499704430,1992633115588943693459627] N = 1936900914619005510

E = ellinit([1,1,1,-30152812499704430,1992633115588943693459627]);
generators:
  [132539377,569561532311]
  [136230497,642713350231]
  [171581555,1367527876091]
  [2041064177,91886986203511]
  [20660191697,2969509496049431]
  [-3416901984423/17161,721062266336463341/2248091]
  [1343074393/9,24391378540397/27]
torsion generators:
  [35540177,982778124311]

8. [1,1,0,-5157621363938497,203250515623581340378681] N = 2011020983594216490

E = ellinit([1,1,0,-5157621363938497,203250515623581340378681]);
generators:
  [-180142737/4,4693566615587/8]
  [-33813563,582242890894]
  [11674022123/4,1260947477060677/8]
  [-23862615697611/354025,104198692671858646247/210644875]
  [-7054497063/14161,764512563774346871/1685159]
  [397523670103/169,250519128231756643/2197]
  [38497791473737/546121,175891707053486554291/403583419]
torsion generators:
  [45110050,249747640609]

9. [1,-1,1,-89370379241164742,10520290211083247769005909] N = 2698227642570635610

E = ellinit([1,-1,1,-89370379241164742,10520290211083247769005909]);
generators:
  [-126912183,4451843256091]
  [69471817,2155623336091]
  [192579087,671876838421]
  [196507717,739173051091]
  [361722217,5051734101691]
  [842087817,23073310056091]
  [195006431817,86113730169336091]
torsion generators:
  [173477817,486995646091]

10. [1,1,1,700175330673221,16719390490661090685920] N = 4272863122202808237

E = ellinit([1,1,1,700175330673221,16719390490661090685920]);
generators:
  [2882906,136972608711]
  [153680135/4,2533863529767/8]
  [-20146726580/3249,20404809902355500/185193]
  [224516837/16,10955988393279/64]
  [29397634195/324,5304024920189825/5832]
  [14875181720/121,1863665946216840/1331]
  [2456321870384/63001,5081760026469428208/15813251]
torsion generators:
  [22569826,209796315511]

Rank 8

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,-1,1,-29136717497212840886,60533108273018762726766844101] 830429816548607909946 48.168475092 146.069195000 129.10328657 [[3670371497,55101761179419]]
2 [1,-1,1,-96024481009434638942,362120228655799988685458307309] 2542465729285305792210 49.287421323 149.647008000 134.12742450 [[4310836217,168176378234691]]
3 [0,1,0,-40653066371981852305,99767215262970013788832370975] 6913619279132804573280 50.287780228 147.068421000 125.84208018 [[3464270315,22567611775500]]
4 [1,1,1,-6769916094364055688,9632928478073408037976861656] 7107840925923552917025 50.315485483 142.393115000 134.25446603 [[1632932220,54149670874152]]
5 [1,0,0,-228454473893195280390,1329069192073918706341388999475] 17261904761372158878705 51.202788989 152.247209000 128.75469217 [[8433449805,47141747567310]]
6 [1,-1,1,-3961248372960718748,3046398107133845961154763606] 18082853740749081030705 51.249251135 140.090660000 127.77645936 [[1150088760,3440073050437]]
7 [1,0,1,-165870106386207383082,797040010380379297766763083200] 19479441722836754745294 51.323646592 151.286813000 141.02473064 [[1432478429,749915344898379]]
8 [1,-1,1,-3218760771095287181417,70287221093926904083268958660009] 25410869169394584077730 51.589463955 160.183453000 142.00690521 [[28664320967,1255165497937116]]
9 [0,-1,0,-194662624315566253780,1037280856440758932964502379972] 42033232163370478906920 52.092747500 151.767002000 140.14052023 [[3276835224,659233660413350]]
10 [1,1,1,15826828814078107235,-6612014528461592389020920845] 42744841527109526140830 52.109535475 144.238318000 136.85539461 [[4455502463,390321839141268]]
generators for these ten

1. [1,-1,1,-29136717497212840886,60533108273018762726766844101] N = 830429816548607909946

E = ellinit([1,-1,1,-29136717497212840886,60533108273018762726766844101]);
generators:
  [-638467179,280848465250983]
  [3140672159,1804178911125]
  [3166266237,4590615769523]
  [14248118079/4,352574749380135/8]
  [-179184831007/121,421810834882290273/1331]
  [30415594609/9,695152370727065/27]
  [109998783148293/37249,112046486866613106851/7189057]
  [183331065552791/58081,50060627137782104445/13997521]
torsion generators:
  [3670371497,55101761179419]

2. [1,-1,1,-96024481009434638942,362120228655799988685458307309] N = 2542465729285305792210

E = ellinit([1,-1,1,-96024481009434638942,362120228655799988685458307309]);
generators:
  [-7938235783,790037265530691]
  [5065687457,75375082920051]
  [5543816669,12695077276587]
  [5592123257,3933317725251]
  [27037665089,4187029708157067]
  [34998349689,6314280344267843]
  [-110641817047/49,258375402074895813/343]
  [1486947843993/289,320849338232201683/4913]
torsion generators:
  [4310836217,168176378234691]

3. [0,1,0,-40653066371981852305,99767215262970013788832370975] N = 6913619279132804573280

E = ellinit([0,1,0,-40653066371981852305,99767215262970013788832370975]);
generators:
  [3676736690,406164886875]
  [3678737315,115276012500]
  [3797207315,12255776032500]
  [40050327185/49,88839868033702500/343]
  [327994438315/121,129640534871490300/1331]
  [640171045115/169,24777076709704740/2197]
  [721083090365/196,34242487459125/2744]
  [106550621894915/32041,210915819716901997500/5735339]
torsion generators:
  [3464270315,22567611775500]

4. [1,1,1,-6769916094364055688,9632928478073408037976861656] N = 7107840925923552917025

E = ellinit([1,1,1,-6769916094364055688,9632928478073408037976861656]);
generators:
  [-3131394160,11267160209367]
  [-2600315810,98258325017717]
  [227934949831/169,119438976869482964/2197]
  [1371356051710/1849,5633104990308653444/79507]
  [4988400305335/1849,8340682468870875644/79507]
  [171133089049435/124609,2379219832253811182434/43986977]
  [14924346015677370/18241441,5307944609064001358472372/77909194511]
  [959611202914621660/506565049,684665723564248482120273656/11401259557843]
torsion generators:
  [1632932220,54149670874152]

5. [1,0,0,-228454473893195280390,1329069192073918706341388999475] N = 17261904761372158878705

E = ellinit([1,0,0,-228454473893195280390,1329069192073918706341388999475]);
generators:
  [8721670755,726343612185]
  [95241484965/16,27159731366842965/64]
  [138245827125/16,889862391103485/64]
  [349252042905/64,253367491715495265/512]
  [424592674230/49,3396701932025835/343]
  [429420245745/49,2062217613341955/343]
  [17569412239830/841,58095097820577529965/24389]
  [2970570585278505/341056,530017744171057259865/199176704]
torsion generators:
  [8433449805,47141747567310]

6. [1,-1,1,-3961248372960718748,3046398107133845961154763606] N = 18082853740749081030705

E = ellinit([1,-1,1,-3961248372960718748,3046398107133845961154763606]);
generators:
  [-165337470,60801572345107]
  [-85442970840/121,98633672055337862/1331]
  [29706619261/16,2927185674805233/64]
  [101711508996/121,23455434039076451/1331]
  [314172219664/225,52026095606770483/3375]
  [434700373815/361,32497214672774668/6859]
  [2244894247440/2209,870617521666124051/103823]
  [5558658434715/2209,9856146926316056546/103823]
torsion generators:
  [1150088760,3440073050437]

7. [1,0,1,-165870106386207383082,797040010380379297766763083200] N = 19479441722836754745294

E = ellinit([1,0,1,-165870106386207383082,797040010380379297766763083200]);
generators:
  [-14271692950,507364417949724]
  [5863018595,161495305451919]
  [-23941054950493/2209,119360523316926264579/103823]
  [-861979603571/144,2168934409983615277/1728]
  [300798899906/49,34536394568359188/343]
  [530282983349/121,522803097532696821/1331]
  [584062752839/121,439140330226730871/1331]
  [1070405325815/121,197368981706623959/1331]
torsion generators:
  [1432478429,749915344898379]

8. [1,-1,1,-3218760771095287181417,70287221093926904083268958660009] N = 25410869169394584077730

E = ellinit([1,-1,1,-3218760771095287181417,70287221093926904083268958660009]);
generators:
  [16046149607,4771789273311516]
  [-17188778380753/289,35199544477887444924/4913]
  [-1589993919889/25,526536241343128044/125]
  [521950564709/16,1997430515394927/64]
  [303439112743/9,8129681790038932/27]
  [40013693056079/1225,276328117844318772/42875]
  [2170049834335423/7569,99243715411668013987948/658503]
  [5218442839383083/310249,790129969690493353338288/172808693]
torsion generators:
  [28664320967,1255165497937116]

9. [0,-1,0,-194662624315566253780,1037280856440758932964502379972] N = 42033232163370478906920

E = ellinit([0,-1,0,-194662624315566253780,1037280856440758932964502379972]);
generators:
  [7031858464,127064451019410]
  [7454031324,20610460791250]
  [984284045792,976421355403401006]
  [-34574415098/9,35496561588671240/27]
  [44728774421/4,4069503744640125/8]
  [88199150416/9,7182726532441750/27]
  [6123640574349/676,2308069546272927575/17576]
  [513505970958896/54289,2600680278276032664270/12649337]
torsion generators:
  [3276835224,659233660413350]

10. [1,1,1,15826828814078107235,-6612014528461592389020920845] N = 42744841527109526140830

E = ellinit([1,1,1,15826828814078107235,-6612014528461592389020920845]);
generators:
  [74148450463,20219620752656268]
  [12806495365/16,5186997082014831/64]
  [991036579117/784,2725411412178003261/21952]
  [256819565663/121,254430626612895468/1331]
  [272022744337/49,172300303625272374/343]
  [781765755527/529,1720863986735709756/12167]
  [7404600057737/1024,22842048561065706699/32768]
  [7532945381069/1936,28745930126024263125/85184]
torsion generators:
  [4455502463,390321839141268]

Rank 9

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [0,1,0,249318358925091505922960,437094728595008754523539867027314900] 536728523406795140805407760 61.547534656 177.654062000 170.21128730 [[581750505170,882620958768080400]]
2 [1,-1,1,-73618642017474515398636907,243125105184756200058212321227222468539] 2108987770780492075994930190 62.916005614 190.296425000 165.96706264 [[4818655044779,518378704095525738]]
3 [1,-1,1,-1272436048402348713084086,548474186580718102061552744510808380] 2401768163536149420691858179 63.046002712 178.122529000 166.42608849 [[1120819589280,728228437471201900]]
4 [0,0,0,-537028752964331311948812,151055073868460837397703542782542384] 3407621442773364069580883520 63.395812035 175.534619000 162.88642018 [[657735739598,287017898332131000]]
5 [1,1,0,-2865810038344413995917296,1964429644286988908697797224491158820] 3682590235087728259825836306 63.473413884 180.659676000 170.86593919 [[988566989472,312211064498668458]]
6 [0,0,0,-1154028987379000568180172,476874822756497046426048134610882736] 3716012842930786646909643840 63.482448788 177.829508000 163.67979177 [[415540427582,262834716304614600]]
7 [1,-1,1,219792557452873548954739,-20768288942605579605232414571054923] 4145713793217534004171727022 63.591872490 172.854516000 165.64211443 [[585692717583,555766092106290232]]
8 [1,1,1,-1841939043304416653141400,962190332674270473405156542715290760] 6830331854767804004039231385 64.091170771 179.232186000 154.17709229 [[801540411258,27659001675621308]]
9 [0,0,0,-92837395950033552980700,11530379178598796179215630517096000] 7205437693496481352981077600 64.144633487 170.383732000 160.70693650 [[178197249770,25407327956649300]]
10 [1,0,0,-74817712498045443069349105,249055725417638127600847825603952905577] 7518548021709644820002553990 64.187170548 190.344894000 174.66597756 [[3849828385994,4251981866432483003]]
generators for these ten

1. [0,1,0,249318358925091505922960,437094728595008754523539867027314900] N = 536728523406795140805407760

E = ellinit([0,1,0,249318358925091505922960,437094728595008754523539867027314900]);
generators:
  [-154272884830,628458393602169600]
  [393684430655,772181081928832290]
  [-100164532185436750/323761,105882294977866038402495120/184220009]
  [85420963109195/529,8443268274387392794920/12167]
  [249151400256875/49,3958155649759637280240/343]
  [5816050327596508/48841,7387863672554917383570354/10793861]
  [11420537353315426562/4068289,39574276406697962095747382208/8205738913]
  [88682486277677681330/595701649,10046957244947204302524525757200/14539290147143]
  [1580709758725045518866/96727225,62878555505961238264875113268864/951312257875]
torsion generators:
  [581750505170,882620958768080400]

2. [1,-1,1,-73618642017474515398636907,243125105184756200058212321227222468539] N = 2108987770780492075994930190

E = ellinit([1,-1,1,-73618642017474515398636907,243125105184756200058212321227222468539]);
generators:
  [698851624325627/121,4332202408810337707686/1331]
  [6299370158046971/3025,1654151718351078793755654/166375]
  [6758551796688371/1369,3293419745959078534914/50653]
  [7755229551031763/2809,1161994554080872655119842/148877]
  [11898383005383659/2401,792291148396733257098/117649]
  [52580899754606331/9409,2280010624034622723221546/912673]
  [288354786617579507/80089,112732964601982151559221886/22665187]
  [1647140965456744091/69169,1992264742064149198734720486/18191447]
  [518033244135787288680203/104634781729,365569647016359161175931777278282/33846526750224817]
torsion generators:
  [4818655044779,518378704095525738]

3. [1,-1,1,-1272436048402348713084086,548474186580718102061552744510808380] N = 2401768163536149420691858179

E = ellinit([1,-1,1,-1272436048402348713084086,548474186580718102061552744510808380]);
generators:
  [-2290989822273/4,8349851907030468983/8]
  [605151835488,8310530636733676]
  [13683287611956,50448894000251952724]
  [-17984990790994509/27889,4886690832472759041129926/4657463]
  [-1445870851058877/1681,69147904042913136559526/68921]
  [201869524527700/289,107561613526196501972/4913]
  [1900399390779600/2401,22690000481046011990995/117649]
  [14184161078304060/25921,528563543018915142202820/4173281]
  [14162137473564642405/23464336,2127348013748059960789140425/113661243584]
torsion generators:
  [1120819589280,728228437471201900]

4. [0,0,0,-537028752964331311948812,151055073868460837397703542782542384] N = 3407621442773364069580883520

E = ellinit([0,0,0,-537028752964331311948812,151055073868460837397703542782542384]);
generators:
  [447579071448,18826237010688700]
  [503121301598,90664706886687000]
  [1521668968238,1690344268013428920]
  [-814605474826732/326041,72676878353797636001250000/186169411]
  [-64794036682562/81,191186944644854969000/729]
  [3640928700382/9,81618793971247000/27]
  [6986556537757/9,12134093760438473125/27]
  [28398199536705/64,5785145736133910431/512]
  [4483512050491892/16129,312354258014147599980000/2048383]
torsion generators:
  [657735739598,287017898332131000]

5. [1,1,0,-2865810038344413995917296,1964429644286988908697797224491158820] N = 3682590235087728259825836306

E = ellinit([1,1,0,-2865810038344413995917296,1964429644286988908697797224491158820]);
generators:
  [-383025968802,1733757958572405504]
  [1037950173966,328766027196977472]
  [4731702660639/4,3833930584977621531/8]
  [-695429277029154607/2070721,5064730877937872594995289171/2979767519]
  [133684452108949/121,509977057101565617698/1331]
  [14480195553534611/18769,1185804063999412349924429/2571353]
  [72326176007424611/12769,18671852361674659717383461/1442897]
  [1476545960608361469/908209,1095643315373821366985227509/865523177]
  [14180442584727801489/2647129,51021030284237192670183634926/4306878883]
torsion generators:
  [988566989472,312211064498668458]

6. [0,0,0,-1154028987379000568180172,476874822756497046426048134610882736] N = 3716012842930786646909643840

E = ellinit([0,0,0,-1154028987379000568180172,476874822756497046426048134610882736]);
generators:
  [-1228249730203,203411567839680585]
  [465995200637,200734707120348555]
  [638827377752,18847576795544820]
  [761137413912,198620355656933420]
  [2812754631197,4414094693347425435]
  [-6639186955232764/7225,535870489507141071035016/614125]
  [-239486334726312/289,4567079779170840855220/4913]
  [141729115619057/196,392781104293493253015/2744]
  [75109015713764849/484,20583903048061848895839465/10648]
torsion generators:
  [415540427582,262834716304614600]

7. [1,-1,1,219792557452873548954739,-20768288942605579605232414571054923] N = 4145713793217534004171727022

E = ellinit([1,-1,1,219792557452873548954739,-20768288942605579605232414571054923]);
generators:
  [124172461155,91861160513326756]
  [2558819437095,4158803485209640408]
  [2361155322765/16,7807141743258902227/64]
  [68419808060925/169,805947481079413498006/2197]
  [15494441189301201/1849,1931678103277911268475908/79507]
  [30817122310006843/841,5410327282976213628505908/24389]
  [122434998154223295/58081,43841187465607730120553016/13997521]
  [134060973317874255/185761,57384965287306036381519496/80062991]
  [5026830767603606991/53919649,9139609820186385991126263272/395931982607]
torsion generators:
  [585692717583,555766092106290232]

8. [1,1,1,-1841939043304416653141400,962190332674270473405156542715290760] N = 6830331854767804004039231385

E = ellinit([1,1,1,-1841939043304416653141400,962190332674270473405156542715290760]);
generators:
  [778362540508,7971950039278808]
  [3134693396407/4,973038995992339/8]
  [783754227058,264405930681308]
  [783808134683,351763491275058]
  [795319249883,18060451124210058]
  [38386941387307/49,76905593334787694/343]
  [38387790771142/49,66716975171893019/343]
  [330717851313352/169,4839219081464178651176/2197]
  [886575038954212/1089,1694144059461122991971/35937]
torsion generators:
  [801540411258,27659001675621308]

9. [0,0,0,-92837395950033552980700,11530379178598796179215630517096000] N = 7205437693496481352981077600

E = ellinit([0,0,0,-92837395950033552980700,11530379178598796179215630517096000]);
generators:
  [-347348335452070/2209,15483385310094705933900/103823]
  [3624742658627840/41209,532040277993637029420600/8365427]
  [5318566365294786/22801,174434782413109694756116/3442951]
  [198705981829026305/824464,42049755833271781505118975/748613312]
  [2548167484946202365/15311569,1573216410009811307846172075/59914169497]
  [6243541952101899970/16818201,11588127061894629663072880700/68971442301]
  [12262989140149571456/58690921,15925276226645880458078455704/449631145781]
  [13756504304527640330/55905529,25028941831041821350962563100/418005640333]
  [19444379410941125505/30118144,77647595341119524325939638975/165288374272]
torsion generators:
  [178197249770,25407327956649300]

10. [1,0,0,-74817712498045443069349105,249055725417638127600847825603952905577] N = 7518548021709644820002553990

E = ellinit([1,0,0,-74817712498045443069349105,249055725417638127600847825603952905577]);
generators:
  [4665576389834,1243535883682841723]
  [5060110297994,180375394768787003]
  [8003931131402,12766164387776793275]
  [12051430246754,33131678314056261683]
  [136228878463994,1586894419898507441003]
  [-259166996165481574/29929,81474035447324570592156151/5177717]
  [1264470588263834/361,37547270629844218789577/6859]
  [7105143772980986/1369,37575530906117158437359/50653]
  [5419735521837622394/1104601,329200121933070212068719953/1160935651]
torsion generators:
  [3849828385994,4251981866432483003]

Rank 10

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,-1,1,-326652986756280439298808750317,71297354488784251332784398142383987320276309] 12441034926946153882022260307768910 78.506308346 215.489691000 203.87170759 [[133852664458767,5474683294732907613016]]
2 [1,-1,1,-118627699282000440895567269581,15312238856253369027772251664973282853020021] 21231627710474670364487625916840938 79.040800012 212.450966000 202.03912509 [[43617312917967,3197029537741315318648]]
3 [0,0,0,-1219617061196330304384542152572,518421489181104191662507254766989215664849536] 27224080399408186552476519408543840 79.289409959 219.441872000 198.66301106 [[680512193578882,1897434527294685725100]]
4 [1,0,0,-27227959822030883481660949639674,54676988991001966689322545553705118390298499044] 50602109704855750262657748221866866 79.909301338 228.758995000 213.20809148 [[2300879854213140,64882148889456960354102]]
5 [1,-1,1,-1243763883354263091668421608561,532688172077316525433436028544237463753530401] 95872894991771433812104612981479282 80.548331372 219.500688000 206.64493978 [[981374118687047,16038925653723255970644]]
6 [0,0,0,-1518426308912492890837664761962,720175870235595998160629433644452843466866709] 172186176309954886778696599644848520 81.133884381 220.099285000 198.29573423 [[748399971521398,1722315865145821958025]]
7 [0,0,0,-3123539885838916490562652730700,2124621830902586944408688768063715547360666000] 187455817312613230634302607218785600 81.218851245 222.263162000 206.13250967 [[810733571833970,11187018368459229891600]]
8 [1,0,0,-8874883833753056057198296069971,10176340399540744124010925860604455083883330465] 439103322830613241474754700030329310 82.070042814 225.395937000 204.96683171 [[1596540707844558,8758829978890633501881]]
9 [1,0,0,-273006478431118488568437133822174,1736187111516655442844466491676088353707715495364] 1186118617225035308071865703374193222 83.063749658 235.674748000 218.33093640 [[11078131145900060,267112462149137045222738]]
10 [1,-1,1,-120389965398027482634502035691631,508418524718676997790711512939977607159051169655] 1764875581069345194499470063902925666 83.461143543 233.218470000 215.97534409 [[5341067449936115,133312556429227016885094]]
generators for these ten

1. [1,-1,1,-326652986756280439298808750317,71297354488784251332784398142383987320276309] N = 12441034926946153882022260307768910

E = ellinit([1,-1,1,-326652986756280439298808750317,71297354488784251332784398142383987320276309]);
generators:
  [14961583349967,8149445233517627397016]
  [776823972263967,16921115704804200093016]
  [857105769190287,20517725152225349774296]
  [939992583429967,24388732787266259253016]
  [1439512375013967,50833334280913319493016]
  [1831090722272967,74917405560588049278016]
  [2562103395993967,126700365977701273973016]
  [2751534231093967,141436213119256039173016]
  [56533477280421967,13441150332264041869125016]
  [858321224143658247/841,690250012202797210556607224/24389]
torsion generators:
  [133852664458767,5474683294732907613016]

2. [1,-1,1,-118627699282000440895567269581,15312238856253369027772251664973282853020021] N = 21231627710474670364487625916840938

E = ellinit([1,-1,1,-118627699282000440895567269581,15312238856253369027772251664973282853020021]);
generators:
  [-162370659524649,5503918597946059842520]
  [-92750752371489,5051449356004927390120]
  [168641533366887,320688792342061404568]
  [240939414916807,846827799037763480088]
  [242383331141103,893758733474348821048]
  [476785915228047,8193746754079314150328]
  [1269911786812303,43733530182843243197368]
  [6564566673785337,531156245432320184322394]
  [7201189343163295/49,351884996618969070417096/343]
  [9911567807853269295/37249,11562439263330195418875784504/7189057]
torsion generators:
  [43617312917967,3197029537741315318648]

3. [0,0,0,-1219617061196330304384542152572,518421489181104191662507254766989215664849536] N = 27224080399408186552476519408543840

E = ellinit([0,0,0,-1219617061196330304384542152572,518421489181104191662507254766989215664849536]);
generators:
  [-817436931196843,31131456689604805313375]
  [638076902334562,1989966773384330220]
  [1020756255429536,18359191695107671469840]
  [1245796219050202,30537078074312839640820]
  [-6275749059570653758/5041,3706841623222811120004039220/357911]
  [-46890291040970603/121,40652306924057457033456225/1331]
  [2802489483902233/4,22408842660996262888035/8]
  [6702140703875338/49,6454993319800306389256140/343]
  [611456865544991002/961,1623849081498558916159500/29791]
  [26402609731261726933/26569,73466124506646865273811656575/4330747]
torsion generators:
  [680512193578882,1897434527294685725100]

4. [1,0,0,-27227959822030883481660949639674,54676988991001966689322545553705118390298499044] N = 50602109704855750262657748221866866

E = ellinit([1,0,0,-27227959822030883481660949639674,54676988991001966689322545553705118390298499044]);
generators:
  [-5776925883678636,138485680604688976341942]
  [-6124932541025889/4,2436772945638885831649977/8]
  [3447519905706132,42227787609941215094454]
  [324391752127437780/529,2378358595273334431475868666/12167]
  [1914033903030140448/361,1671146208230278509236958834/6859]
  [47630973783088585524/15625,3527103343742079513168493518/1953125]
  [108634665333670341204/36481,11166811115984622165701488074/6967871]
  [2331989675564255450580/877969,27223925396339184547938035821446/822656953]
  [828899126583462980667732/162384049,456633535074678115030754188319657562/2069259936407]
  [1406962787432949478072308/419881081,280679220988339525465586039679882306/8603783230771]
torsion generators:
  [2300879854213140,64882148889456960354102]

5. [1,-1,1,-1243763883354263091668421608561,532688172077316525433436028544237463753530401] N = 95872894991771433812104612981479282

E = ellinit([1,-1,1,-1243763883354263091668421608561,532688172077316525433436028544237463753530401]);
generators:
  [-721122679420513,32474571970896280104564]
  [676534503526967,942396679375966862004]
  [689781376045847,1720420381425131473044]
  [-1116499359109385389561/1203409,39339637971987601568404710127860/1320139673]
  [-3954145313922681/25,3366777851546307588571812/125]
  [730610578829108231/625,406301702175705239745659796/15625]
  [886791538033151759/1849,542115465313604186193114540/79507]
  [145238038976986662779/237169,95243017038484714043532135660/115501303]
  [1173820448894283788903/1745041,1489481265745030271069768783988/2305199161]
  [2149845742600104296091/3214849,205070003524033720930962100660/5764224257]
torsion generators:
  [981374118687047,16038925653723255970644]

6. [0,0,0,-1518426308912492890837664761962,720175870235595998160629433644452843466866709] N = 172186176309954886778696599644848520

E = ellinit([0,0,0,-1518426308912492890837664761962,720175870235595998160629433644452843466866709]);
generators:
  [-825110947090247,37567324159599876037110]
  [-321734683300562,34284143407089487452255]
  [251272242256153,18828230090132452184910]
  [710920229087173,18915847531589385000]
  [721473473611138,464548931739172394595]
  [936517883952658,10932988333851045777645]
  [-277273875475751207/256,135489746744220032632980765/4096]
  [6422721546455002/9,2716260595248925494055/27]
  [14462044121030497/4,1649086239767378700884895/8]
  [659801767248118585/961,34009992844212199882633722/29791]
torsion generators:
  [748399971521398,1722315865145821958025]

7. [0,0,0,-3123539885838916490562652730700,2124621830902586944408688768063715547360666000] N = 187455817312613230634302607218785600

E = ellinit([0,0,0,-3123539885838916490562652730700,2124621830902586944408688768063715547360666000]);
generators:
  [-1950332338601230,28247050862142209168400]
  [-164349407766190,51317975058955534465200]
  [1007861884074245,544612158496118690775]
  [1033634668656845,598870052946853870275]
  [177987925635835520/361,182046805675652994234687900/6859]
  [2158271962342471810/2601,1353414578148616055548193200/132651]
  [37529110310662240370/36481,1258326583654763301729296400/6967871]
  [46068797956252024640/18769,247243711779369533077512686100/2571353]
  [62474820767094228770/121801,1089932630686338004828193811600/42508549]
  [2172885329090136251105/2184484,4358478099884026353677087122575/3228667352]
torsion generators:
  [810733571833970,11187018368459229891600]

8. [1,0,0,-8874883833753056057198296069971,10176340399540744124010925860604455083883330465] N = 439103322830613241474754700030329310

E = ellinit([1,0,0,-8874883833753056057198296069971,10176340399540744124010925860604455083883330465]);
generators:
  [-3318834660586482,55450569401122281242361]
  [501623683987158,76489935019725329679081]
  [1718075432936814,81866711288971561785]
  [1727489152760334,529531364521868124921]
  [1812900015784718,6734408760756053357561]
  [1912367183605368,14075299635529954796961]
  [207673221369130368/121,318710634725105037227811/1331]
  [18213625617561267054/10609,217747983301595875367619135/1092727]
  [20556985623132491358/11881,946007960590743464001502509/1295029]
  [15102104616017821344012/5139289,1133890240332659347569912295076193/11650768163]
torsion generators:
  [1596540707844558,8758829978890633501881]

9. [1,0,0,-273006478431118488568437133822174,1736187111516655442844466491676088353707715495364] N = 1186118617225035308071865703374193222

E = ellinit([1,0,0,-273006478431118488568437133822174,1736187111516655442844466491676088353707715495364]);
generators:
  [8607539605009244,154930272280096254710354]
  [9639309290785724,15557094353622625246322]
  [-579945812851023997444/38809,12037074589857637873468805682826/7645373]
  [-11852960355108356/25,170729955564292867457815562/125]
  [156309771247786157/16,2462233808275341587596331/64]
  [256448184629339516/25,15359061594512099920270474/125]
  [3225326722355929724/121,4871966538682523397282649174/1331]
  [15936473158648576412/1681,515266412817353863719660994/68921]
  [226077429302724502879772/24196561,3855573075420337094654130429523966/119022883559]
  [4904515920036014485125860636/313467025,343474089346074181393035382592919659619394/5549933677625]
torsion generators:
  [11078131145900060,267112462149137045222738]

10. [1,-1,1,-120389965398027482634502035691631,508418524718676997790711512939977607159051169655] N = 1764875581069345194499470063902925666

E = ellinit([1,-1,1,-120389965398027482634502035691631,508418524718676997790711512939977607159051169655]);
generators:
  [-10843781993396455,734034526595533334764644]
  [5767036523837915,77002555592284421037990]
  [6110508753925473,30507398849930577680156]
  [6149159863695603,25188116292633799066470]
  [6276740066727755,7046825720139384478350]
  [7212891886772469,123754828056823599216632]
  [9733697869963055,508720896394681796031894]
  [20909472858771251,2670744446867549360454054]
  [259402251243146675,132001265113459677404689830]
  [94826944986880919/25,40762784544521665316391762/125]
torsion generators:
  [5341067449936115,133312556429227016885094]