Average Ranks

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Description

This graph shows the average rank functions for four different sets of elliptic curves up to conductor 100000000. The highest is the graph of the function of the conductor N showing the average rank of all squarefree optimal curves up to conductor N, the second one is the graph of the function showing the average rank of all squarefree-conductor curves up to a given conductor. Next is the average rank for all optimal curves up to N, and the last is for average ranks of all curves up to conductor N. Each graph gives the pairs (N,r), where r is the average rank for abovementioned curves of conductor < N and N runs through the multiples of 100000.
Conductor NAverage Rank of all curves with conductor <= NAverage Rank of optimal curves with conductor <= NAverage Rank of optimal curves with squarefree conductor <= NAverage Rank of all curves with squarefree conductor <= N
1000000.6985880237890.7417719033280.8406954570950.774544043461
2000000.7260259374790.7669202468820.8723455913940.810544128188
3000000.7410054174520.7802200551180.8878303653450.828980943947
4000000.7513599382450.7893740119890.897599209250.841069671531
5000000.7587146695790.7958466226860.9046395783250.849761615345
6000000.7645247274650.8009485537380.9101323479070.856675955551
7000000.7698651432020.8056956811180.9150993082690.862556523657
8000000.7739744163870.8092906034930.9193277942330.867531533186
9000000.7772732440830.8123726210860.9228285478130.871771500149
10000000.7805201419320.8151726477410.9264200849310.87604956366
11000000.7834735067840.8177540381920.929176775820.879482188842
12000000.7860427861010.8200453171010.9317085496720.882579946238
13000000.7882186099570.8218833208990.9336171979090.885156135162
14000000.7902463781230.8236479867610.9355663955270.88754589229
15000000.7922092347510.8252666957390.937372834680.889871729774
16000000.7940575640620.8268605207010.9391311449930.892072134424
17000000.7957840944950.828399795830.9408194053170.894063018586
18000000.7972672744040.8296681377370.9423983836930.896033669259
19000000.7985377088140.8307144873680.9436069511720.89765907388
20000000.7999044784470.8319115014930.9448204040560.899169264865
21000000.8013210717750.833150848080.9459889761080.900637005609
22000000.8025558639440.8342229483660.9472228379860.902184518222
23000000.8037138928730.8351691275620.9483517617690.903584881091
24000000.8048290757080.8361129166550.9493072546730.904866778396
25000000.8058476334580.8369972273220.9501238417540.906011584713
26000000.8067161696170.837729632520.9508708812890.907012142431
27000000.807687963560.8385283066690.9516894226570.908109076085
28000000.8086040530210.8392859519720.9522833402760.909033761003
29000000.8095278322590.8400783081710.9530362541630.910032532744
30000000.8103731160080.8407887648130.9537921672090.910984802476
31000000.8111105584860.8414220572490.9544558157360.911892035858
32000000.8119067618630.8420954682080.9551823605440.912845953903
33000000.8126865603380.8427815923170.9559371701090.913793511952
34000000.8133836022780.8433786334620.9565365424350.914595762237
35000000.8140164524390.8439242534070.9571944966190.915440607639
36000000.8147201014650.8445234223020.9578487953020.916336718205
37000000.8153167117630.8450239009520.9584287857980.917137933686
38000000.8159993837410.845619942990.9590326643370.917913190096
39000000.8166397149010.8461766513740.9595789770720.918659234404
40000000.8172116448060.8466725337160.960035729920.919314251487
41000000.8178183486540.8472121677440.9606057364820.920022915659
42000000.8183156343750.8476341102710.9610901286960.920629279316
43000000.8188173390270.8480559319790.9615783535540.921287110714
44000000.8193737276650.8485314661780.9621065013680.922018623879
45000000.8199247734730.8490095473890.9626917362140.922802662944
46000000.8204000218740.849410616550.963111069610.92333787182
47000000.82085113890.8498087153560.9634745605380.923864420467
48000000.8213469861010.8502216082880.9641200855230.924625306132
49000000.8217575603510.8505701563860.9646525320960.925270170361
50000000.8221563056670.8509083336110.964917853950.925641948444
51000000.8225929538440.8512676553960.9652177862310.926087987437
52000000.8230527560910.8516474227340.9655551869910.926550943953
53000000.8234928187320.8520143871960.965916312510.927049133168
54000000.8239201402520.8523537353580.9662570788880.927535191773
55000000.8243105336410.8526649103010.9665590924990.9279705839
56000000.8247108798980.8529874440260.966839535730.928382453721
57000000.8251161057140.8533297365330.9671187011730.928783195832
58000000.8254958273060.853660381430.96750132350.929277189159
59000000.8258656760370.8539766135250.9678274121750.929717066447
60000000.8262372516680.8542955389520.9681596765080.930171442364
61000000.8265950620170.8545805780070.968456145120.930612042572
62000000.8269237559090.8548660445140.9687035454050.930951031655
63000000.827287835910.8551781547870.968991074890.931324460342
64000000.8275978914810.8554362719050.9692311309190.931656713719
65000000.8278777913780.8556720553230.9694871677670.932013127423
66000000.8281873841620.8559288665090.9697845079770.932390436483
67000000.828478152710.8561864021920.9700430763370.932743471375
68000000.8288216640740.8564772114040.970317803760.933122311601
69000000.8291346584140.8567472365110.9706596884170.933539735538
70000000.8294337828950.8569952053650.9709201712330.933917819148
71000000.8297761822950.8572815334370.9711561335910.934252182063
72000000.8300843323970.8575430662220.9714295983690.934599800285
73000000.8303704638630.8578004544020.971747269450.935009604289
74000000.8306211281750.8579910431080.9719564181380.935333381474
75000000.8308785450820.8581981116740.972211908070.935670718696
76000000.8311519885890.8584166918710.9723758213710.935938048374
77000000.8314276927410.85864434440.9726096597830.936276109984
78000000.8316825554530.8588498225540.9727810580270.936523572423
79000000.8319701281280.8590894632780.9730235297940.936854198237
80000000.8321940602790.859285338940.973227579720.93713958948
81000000.8324506901120.8595061369630.9734194596840.937415795301
82000000.8327044842410.8597121775760.9736563752360.93774758072
83000000.8329411138490.8599106911650.973771355460.937961054292
84000000.8331580823560.8600792671910.9739751148340.938238616609
85000000.8333816937490.8602595795820.9741293040960.938487064754
86000000.8335757082730.8604050955210.9742072562370.938650244265
87000000.8338294119370.8606150502550.9743921179480.93891636842
88000000.8340781889590.8608193558560.9746086535630.939219224156
89000000.8342950035440.8610081349850.9746922045140.93938279974
90000000.8344684807480.8611475514670.9748413026290.939599489816
91000000.8346713630260.8613048072570.9750059504730.939868016354
92000000.8348665222010.8614724050740.9751657212830.940087020243
93000000.8350681546650.8616327283580.9753400741350.940336961913
94000000.8352729062990.8618118591910.9754718761910.940535046234
95000000.8354575870170.8619556109350.9755868078210.940733777829
96000000.8356966096730.8621444881540.975824847470.941043355185
97000000.835856465610.8622846480310.9759465573970.94124975402
98000000.8360554077230.8624575406630.9761091160450.941487422703
99000000.8362430548030.8626116450420.9762560850610.941695279446
100000000.8364520692010.8627810105190.9764272657970.941934982207
101000000.8366113105470.8629157268510.9766082221990.942163734015
102000000.8367953140830.8630740976330.9768056617320.942427095391
103000000.8370050728660.8632493797140.9770306221670.942715324478
104000000.8371487445410.8633499864950.9770183427790.942793857906
105000000.8373267303160.8635048793070.9771831992380.943018733288
106000000.8375132429130.8636697932560.9773346853310.94322308666
107000000.8376779101030.8637917632330.9774586357250.943414851739
108000000.8378473112530.8639340125110.9776238720150.943654308165
109000000.8380083534880.8640672617220.977705694480.943786755004
110000000.8381775157020.8641981834790.9778672823980.944027018996
111000000.8383306126450.8643254682420.9780170415130.944241930814
112000000.8384962995370.8644604444490.9781269096040.944404420331
113000000.8386806050130.8646083794330.9782478271870.944586346899
114000000.8388437627310.8647427599390.9783947316190.944790381036
115000000.8390213977410.864895046830.9785459029810.944997130933
116000000.8391462866810.864993951210.9786310204430.945144516735
117000000.8393080768310.8651309216970.978728141680.945283691962
118000000.8394445344820.8652461208630.9788041264490.945414622538
119000000.8396179127750.8653893606530.978947969060.945621553826
120000000.8397728251620.8655213864710.9790933780940.945814826655
121000000.8399269879760.8656419235340.9791597567320.94594624383
122000000.8400680687570.8657619084050.9792833567420.946130527854
123000000.8402213984450.865889458880.9793867332990.94629729775
124000000.8403613923540.8660063295040.9794651499720.946429534737
125000000.8404938606370.8660973166540.9794735716230.946500024326
126000000.8406224437160.8662078478130.9795817270570.946666743322
127000000.840744263230.8662988322450.9796647896140.946803943675
128000000.8408776490490.8664017297550.9797682150670.946965231955
129000000.8410133513830.8665119578290.9799113925170.947156874842
130000000.8411801813930.8666487396770.9800173235460.947315744776
131000000.8413406099250.8667809216860.98008515160.947439814812
132000000.8414388479230.8668561060610.9801438728420.947540019812
133000000.8415302354410.8669207797990.9802193850030.94768440302
134000000.8416588915810.8670343733250.9802992113430.947810534518
135000000.8417843755490.8671334737630.9803891041040.947953655162
136000000.8418985521690.8672368347110.9804818508370.948079952931
137000000.8420324749350.8673481105450.980603901960.94825403819
138000000.8421681966610.867456514750.9806938180670.948383263716
139000000.8423173744080.8675800783550.9807846729870.948531741904
140000000.8424536988190.867693198290.9808534369160.948643019849
141000000.8425797007150.8678045997370.9809609500770.948793743433
142000000.8427015645540.867906140080.9810831015360.948970426932
143000000.8428282543130.8680076592820.9811675984130.94911183556
144000000.8429227363080.8680778516670.9812906813880.94928156266
145000000.843046495140.8681812841370.9814036830280.949432976875
146000000.8431782838110.8682896575170.9815167271050.949591982402
147000000.843310087120.868393432490.981622224970.949747568156
148000000.8434359253670.8684967521450.9817083339460.949871851259
149000000.8435726315860.8686044845640.9817998954390.950002795055
150000000.8436866782120.8687002581350.9819097752850.950151971536
151000000.8438005280740.868798893020.9819654542230.950245760464
152000000.8439107109520.8688835126040.9820108190330.950348901717
153000000.8440327021490.8689807173730.9820648168210.950461280026
154000000.8441560683880.8690862798170.9821777864890.950616391588
155000000.8442505369570.869152787790.9822123864980.950692373839
156000000.8443732162930.8692477384530.9823232144630.950849852912
157000000.8444813231750.8693372486010.9824202859190.950999655922
158000000.8445893254670.8694251659730.9824837947520.951098041252
159000000.8446975233440.8695118780410.9825487157630.951201601453
160000000.8448178415350.8696184744340.982622301290.951306970409
161000000.8449019241640.869682603380.9827191228080.951440574773
162000000.8450143019410.8697768695590.9828030785280.951557914375
163000000.8451225864570.8698641780210.9829185541650.951715162048
164000000.8452473950520.869971865120.9830367536060.951861545693
165000000.8453513407010.8700570284840.9830978380150.951973361273
166000000.845444594640.8701258244530.9831562321960.95208534402
167000000.8455697250440.8702317855230.9832196301890.952194799203
168000000.845675106830.8703120770350.9832858814070.952300356184
169000000.845782340070.870400420070.9833798704360.952436170298
170000000.8458731761440.8704710717180.9834532315790.952552822871
171000000.8459548529320.8705413237790.9834919268960.952621819663
172000000.8460316906110.8706028978910.9835324448030.952689665695
173000000.8461493579170.870699983870.983617952470.952803464629
174000000.8462387680.8707699857980.9836727629480.95290883172
175000000.8463350707940.8708418102420.9837196112810.952989774453
176000000.8464274262060.8709205402480.9837708689170.95308407581
177000000.8465281400970.8709989071510.9838201337730.953168172515
178000000.8466136245370.8710739997010.9838701810620.953246783399
179000000.8467003994650.8711419953040.983921680480.953330315785
180000000.8467868233010.8712119352780.9840137928530.953445635105
181000000.8468658745160.8712775914310.9840579440060.953525817265
182000000.8469667585450.8713597155170.9841296687810.953631341175
183000000.8470633021960.8714329934210.9841615700160.953703259682
184000000.8471579732180.8715070018050.9842037615250.953784983782
185000000.8472531474620.8715877250880.9842821230320.953886021442
186000000.8473527143890.8716713709370.9843484858250.953984608884
187000000.8474454154960.8717519138430.9844245756490.954099401808
188000000.8475285520990.8718199595520.9844815917470.954197577803
189000000.847601449010.8718747929040.9845197454280.954277310853
190000000.8476759447810.8719330492080.9845537494550.954348137321
191000000.847774034330.8720153091640.9846382234740.954461269425
192000000.8478600959490.8720868993710.9847268134830.954583898634
193000000.847959876990.8721711346640.9848028661580.954690591436
194000000.8480598667340.8722522558660.9848430714280.954779151661
195000000.8481645117410.8723343033890.9848942417290.954867311832
196000000.848257113760.8724027707710.9849632158180.954978585215
197000000.8483322218950.8724633076070.9850016011940.955047618993
198000000.8484128182390.8725277127430.9850840690250.955166782502
199000000.8484933719760.8725971411920.9851430737660.955253660896
200000000.8485939720540.872680199130.9851865840160.955324183221
201000000.8486753553610.8727465013420.9852654567570.955435994717
202000000.8487688812940.8728193130810.985351216160.95555322953
203000000.8488588219090.8728968601370.985422991350.955661767649
204000000.8489545769150.872973580490.9855220354650.955788317534
205000000.8490026426990.8730009995760.9855364594440.955840189889
206000000.8490813884920.8730600647870.9855440841020.955879083621
207000000.8491581917550.8731273743220.9856372697960.955996963918
208000000.8492322822120.8731799594460.9856724381950.956057785883
209000000.8493126353040.8732514365620.9857196715370.956131524671
210000000.8493978140530.8733234269990.9857853261460.956229824753
211000000.8494883067410.8733981047360.9858351369460.956311840361
212000000.849566824760.8734670877250.9858897021930.956396043614
213000000.8496462223310.8735358802880.9859465020510.956479169453
214000000.8497034909880.8735824198990.9860087374390.956574479528
215000000.8497679240740.8736378024230.9860578627390.956649784956
216000000.849838288640.8736992348030.9860799609610.956693517034
217000000.849912789480.8737572920550.9861486539420.956793320152
218000000.8499946864650.8738280150760.9861883062370.956861195252
219000000.8500693497510.8738921146510.9862515820140.956954325512
220000000.8501549300620.8739645692430.9862989664970.957023839533
221000000.850228975430.874024027730.9863460279060.957101013061
222000000.8503025362390.8740834563460.9863902902910.957169212231
223000000.8503844025030.874147111110.9864354718640.957241537485
224000000.8504445069270.8741956653350.986452833410.957285381672
225000000.8505070530340.8742415819360.986489925230.957349123047
226000000.8505801365220.8742987809670.9865102287940.957405378361
227000000.8506289303040.8743421395210.9865497927250.957464833425
228000000.8507187528630.8744150696730.9866000811340.957539202138
229000000.850787873940.8744657672630.9866495240320.957614004624
230000000.8508425925890.8745076451540.9866777045940.957662753978
231000000.8509121268970.8745644429010.986713553510.957730480262
232000000.8509653397130.8746051227570.986727537820.957769763281
233000000.8510318946140.8746586330270.9867483993660.957804170764
234000000.8510883938450.8747043876170.9867933831130.957876182487
235000000.8511432435770.8747488468560.9868175575380.957927254454
236000000.8512088841720.8747973575290.9868593181620.957997262521
237000000.8512752011460.8748516951420.9869171696470.958079238372
238000000.85134932740.8749064287150.9869524022180.958141808922
239000000.8514015070580.8749464031220.9869913850130.95820966911
240000000.8514730462860.8750065005310.9870521085360.958295129515
241000000.851543761450.8750665253690.98707765060.958349787249
242000000.8516103584340.8751212064560.9870915947780.958396021642
243000000.8516620920750.8751570189670.987100559630.958433614897
244000000.8517273904010.8752090992970.9871684989070.958524094974
245000000.8517767039980.87524231770.9871866541570.958568343089
246000000.8518367198640.8752966326390.9872364529170.958643671944
247000000.8519017597680.8753542413570.9872443552120.958676381219
248000000.8519523865220.8753959281020.987280669970.958734103482
249000000.8520206591320.8754427186540.987284900790.958772472783
250000000.8520896082760.8755010764760.9873359631130.958846749416
251000000.8521498113370.8755507171040.9874088223160.958933110819
252000000.852194147860.8755796005660.9874227918750.958972570602
253000000.8522307364890.8756073625890.9874590218640.959037588469
254000000.8522908905320.8756501918710.9874975215190.959097980451
255000000.8523344012750.8756829511830.9875156284570.959138439785
256000000.852380503670.8757191443230.9875568493980.959202419093
257000000.852420951130.8757435207110.9875902583930.959261297392
258000000.8524608712020.875769593830.9876137061590.959308824821
259000000.8525177997040.87581534610.9876627880820.959375836205
260000000.8525762682920.8758611686960.9877372436480.959471762974
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