History of Real periods of elliptic curves over $\mathbb{Q}$ of rank $3$

+Title: Real periods of elliptic curves over $\mathbb{Q}$ of rank $3$+Definition: The real period $\omega_1$ of an elliptic curve over $\mathbb{Q}$ can+  be defined as $\Omega_E$ if $E(\mathbb{R})$ is connected and $\Omega_E/2$ if $E(\mathbb{R})$+  has two components, where $\Omega_E = \int_{E(\mathbb{R})} |\omega|$, $E(\mathbb{R})$+  is the set of real points of $E$,  and $\omega$ is the invariant differential $dx/(2y+a_1+  x+a_3)$ of a globally minimal model of $E$. This table lists $\omega_1$ for elliptic+  curves over $\mathbb{Q}$ of rank $3$ for the first few conductors.+Parameters:+  N:+    title: conductor of $E$+    type: Z+  c4:+    display: $c_4$+    title: invariant $c_4$ of a minimal model of $E$+    type: Z+  c6:+    display: $c_6$+    title: invariant $c_6$ of a minimal model of $E$+    type: Z+Comments:+  comment-Omega: Note that some authors refer to $\Omega$ as the real period.+  comment-period-lattice: The period lattice $\Lambda$ of $E$ is  the lattice of periods+    of $\omega$. In terms of $\Lambda$, $\omega_1$ is the smallest positive  real+    number in $\Lambda$.+  comment-BSD: The Birch and Swinnerton-Dyer conjecture CITE{WikiBSD} states that+    $\frac{L^{(r)}(E,1)}{r!} = \frac{|\text{Sha}(E)|\Omega_E R_E \prod_{p|N} c_p}{|E_{\text{Tor}}|^2}$.+  comment-redundancy-of-N: The conductor $N$ can be deduced from $c_4$ and $c_6$.+    We display it for convenience.+  comment-E-from-c4c6: 'A model of $E/\mathbb{Q}$ is given by $E: y^2 = x^3 - 27 c_4+    x - 54 c_6$.'+Formulas: {}+Programs:+  program-sage:+    language: Sage+    code: "numbers = [E.period_lattice().real_period() \n                        \+      \ for E in cremona_curves([1..10000])\n                         if E.rank()\+      \ == 3]"+References: {}+Links:+  LMFDB:+    title: 'LMFDB: Elliptic curves over Q'+    url: https://www.lmfdb.org/EllipticCurve/Q/+  WikiBSD:+    title: 'Wikipedia: Birch and Swinnerton-Dyer conjecture'+    url: https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture+Similar tables: ''+Keywords: ''+Tags:+- number theory+- elliptic curves+- period+Data properties:+  type: R+  complete: 'no'+  sources:+  - CITE{LMFDB} (list of curves with bounded conductor)+Display properties:+  number-header: $\omega_1$+  group parameters:+  - - N+  - - c4+    - c6+Numbers:+  '5077':+    336, -5400:+      number: '2.075843991543466524942087841753643149885814128284888052596870827360594215192217462023211391604654285'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5077a1}[Curve in LMFDB]+  '11197':+    273, 999:+      number: '1.653335536933760110265560695395294897309748727569401960052043346678684458763199624814897266604466723'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/11197a1}[Curve in LMFDB]+  '11642':+    777, -20709:+      number: '1.875704082435827098288217159897846012929827417086345974229415960353779800762182362370503872390227910'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/11642a1}[Curve in LMFDB]+  '12279':+    496, -7640:+      number: '1.695454941526035732006967079877117358323163564944568491130390575002180461706353466092162571128278935'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/12279a1}[Curve in LMFDB]+  '13766':+    1081, -38125:+      number: '3.265608124182381308349493464664603027468675681760000807375256195243657490438085847904389617679171222'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/13766a1}[Curve in LMFDB]+  '16811':+    48, -5400:+      number: '3.194003850461162784947394782052063472018342397969740844130937674170747120166945689819587736335129879'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/16811a1}[Curve in LMFDB]+  '18097':+    481, -8945:+      number: '1.854053836952336672288298346695970411854856401307168670444932139027377298153534341369830325963346943'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/18097b1}[Curve in LMFDB]+  '18562':+    937, -27541:+      number: '1.807317069348157359478187112983105958118477403978640614206747268354579051939217735742664165340337728'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/18562c1}[Curve in LMFDB]+  '18745':+    7024, -591832:+      number: '2.618927286333791175095089762648521552400000387208541702970606273698313323549585245921718072441816937'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/18745a1}[Curve in LMFDB]+  '20888':+    2496, -86400:+      number: '1.132540774852863115598234995837969492114318556228982325355698565023034923347372767783415812143699889'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/20888a1}[Curve in LMFDB]+  '21443':+    241, -7145:+      number: '3.434848415077769710146263982940223681684697012739618709792183534969211826754169745984280447186070240'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/21443a1}[Curve in LMFDB]+  '21858':+    1561, -63485:+      number: '3.270758461590460975118282828748778044170607572644710150428061756451881403706296582678336830226917861'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/21858a1}[Curve in LMFDB]+  '22481':+    -303, -3321:+      number: '2.457390758366876165046324061677089774201043070337182131959196368463106856779757904738243027724090112'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/22481a1}[Curve in LMFDB]+  '22696':+    5056, -357760:+      number: '1.433659600400976213283683515984810931473889064600804388455192443506020580323194897450042407671112189'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/22696a1}[Curve in LMFDB]+  '24301':+    2113, -97345:+      number: '3.796280092473942812358064080094634854089669303667876612385640800205807086833163330077152841018568905'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/24301a1}[Curve in LMFDB]+  '24546':+    1897, -84149:+      number: '3.256604879434541089117513887288106169875018602605084723867077216126873776623801915882495756121114584'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/24546a1}[Curve in LMFDB]+  '24646':+    601, -11485:+      number: '1.680998242220892276202515025442346460922705414465804972994925002792838881830199730982558305547324249'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/24646b1}[Curve in LMFDB]+  '25071':+    1168, 20584:+      number: '1.116559940103587668949684105546955810713494683962932526596544684931263226941196573842789032024157441'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25071a1}[Curve in LMFDB]+  '25383':+    1552, -60056:+      number: '1.720596888844160797316407851345770481276847090707299619403038080847303964084191601911596392715015269'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25383a1}[Curve in LMFDB]+  '25451':+    2593, -137105:+      number: '2.805376767345587792491786585641147634624388843844733610255426157342804884730185466955375079839205081'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25451a1}[Curve in LMFDB]+  '25751':+    1873, -84025:+      number: '3.057673958679663299846857510538152370266196002668883725440560486626449516042240032797600044809494188'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/25751a1}[Curve in LMFDB]+  '26171':+    2352, -114264:+      number: '3.766959541087013639221355733868826140008946588952348960488663073612594302964523918127901037867718610'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26171a1}[Curve in LMFDB]+  '26198':+    777, -10341:+      number: '1.420899897567113352516808446324645580661589634547820643333447421179629838630069663166848161847438525'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26198a1}[Curve in LMFDB]+  '26284':+    448, -16480:+      number: '3.035208094484576365819069137253472946578198421261724060205152097474538882630424728800907193850294427'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26284a1}[Curve in LMFDB]+  '26743':+    -287, 4751:+      number: '2.023924712358822618917239031439724410577090592367196873049189003653889130230056921897927639234052901'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/26743a1}[Curve in LMFDB]+  '27262':+    11361, -1128465:+      number: '0.917130855987737438951792609406576879764058369999810509007438830118094693891134647485723890028483429'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27262c1}[Curve in LMFDB]+  '27382':+    457, -901:+      number: '1.511803426985222177725891626370460288067977202091138204895861993025155721632166969851750433307292615'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27382a1}[Curve in LMFDB]+  '27448':+    3792, -236736:+      number: '2.813079838059818965621990288043057665416552482528853879304460793476811137492590163390531552570160628'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27448d1}[Curve in LMFDB]+  '27584':+    192, -55296:+      number: '2.157562493776799501917405525117477543267522633402189075601752544232685577309314567043461563169827707'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27584bd1}[Curve in LMFDB]+  '27746':+    553, -23509:+      number: '2.841178471635987442879407359778892484799041792831826795404305174639071101336244866173130721899676869'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27746a1}[Curve in LMFDB]+  '27747':+    15696, -1975320:+      number: '2.172961686271171140917442072976117191094577391552826715075451635913216169226229152174150957085688886'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27747c1}[Curve in LMFDB]+  '27773':+    993, -32049:+      number: '3.731536610080869329785699638402279565278104832568046439255652416521149579652969028221547745587119900'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27773a1}[Curve in LMFDB]+  '27808':+    17472, -2294784:+      number: '1.028248440055731521826561321571597056483223017910482220386339253232930350818170413917072008631048202'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/27808a1}[Curve in LMFDB]+  '28042':+    921, -38205:+      number: '2.796857240896887145400618495329419022488218109935882205942869191014599115839532041972068143577945164'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/28042b1}[Curve in LMFDB]+  '28498':+    633, -18765:+      number: '3.368067869067191918352691861702479995064524741104759265286684063365775542013540685638702795764302215'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/28498a1}[Curve in LMFDB]+  '28571':+    208, -7640:+      number: '3.279674898773183013966254165374661814666999095870961171791444240639722837279159188972025355546381352'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/28571a1}[Curve in LMFDB]+  '29157':+    11536, -1097560:+      number: '0.864748762188697455512273675427258184996822058907733634344562145764657303309805690860459085527064748'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/29157b1}[Curve in LMFDB]+  '30064':+    7312, -627904:+      number: '2.643380760068737387435371184583561676298060143738752065874593387865995137372936600087327219608624143'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30064c1}[Curve in LMFDB]+  '30148':+    1776, -73440:+      number: '1.655029110901514599410395143911589957481125347680730733295904038297002704635297340429832948776340257'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30148b1}[Curve in LMFDB]+  '30376':+    1216, 10880:+      number: '1.133750761955206060617593206758791933153441114171821015616784415316719907054366305551919338319273992'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30376a1}[Curve in LMFDB]+  '30446':+    1897, 36811:+      number: '0.995457313643036361188780787229191025961338821426007024203630246036244356998230560194880522643416234'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30446a1}[Curve in LMFDB]+  '30487':+    1344, 12744:+      number: '1.105491725182012449680818465316372299957063767085193855118743286630426698426729719788177897797959127'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30487a1}[Curve in LMFDB]+  '30767':+    1488, -52056:+      number: '1.475135563401352042668641426198157666672421872340091809958185242571821657661615475973527586419050610'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30767d1}[Curve in LMFDB]+  '30815':+    784, -27352:+      number: '3.067585117729036253583015561060262593709455422420133855374730649729555596367780052997276993434884613'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/30815a1}[Curve in LMFDB]+  '31478':+    217, -10909:+      number: '3.016403883642831564686027667109810674291701755117467479185051649932616303034940900479270603330018295'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/31478b1}[Curve in LMFDB]+  '31737':+    1648, -68120:+      number: '3.377608764975393298490001628302704736967985802539380947520377927031002758921462612182220769083280042'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/31737a1}[Curve in LMFDB]+  '31814':+    -119, -10405:+      number: '2.641537673282419790673527824479842849439393386284456059991650726786666302711693713225035614857529773'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/31814a1}[Curve in LMFDB]+  '32192':+    2496, -138240:+      number: '2.533600042943272288620095417116693913566439541520269285699750104921862316547339239349399845443708823'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32192e1}[Curve in LMFDB]+  '32244':+    1456, -61280:+      number: '2.916474393503243030669095928051333118735054897029521616158992877000582923336056127549015565311477507'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32244c1}[Curve in LMFDB]+  '32276':+    832, -18784:+      number: '1.552357339773422860598255111928360304163925310934180456598502100010990071881743313999698960553761784'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32276b1}[Curve in LMFDB]+  '32336':+    1552, -12736:+      number: '1.133669507382139586615381801656887912698154457999936051265855637752486040315581188978866770304854396'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/32336b1}[Curve in LMFDB]+  '33424':+    14992, -1831616:+      number: '1.164327427613746499338374530989015135492835031217533083738630710600115889156748548133622840700921622'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33424o1}[Curve in LMFDB]+  '33428':+    2176, -81280:+      number: '1.232940477184129582565246741080142452639442019869672935847704863208135614370332875684759631156018612'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33428a1}[Curve in LMFDB]+  '33509':+    1072, -34264:+      number: '1.836612582900331039016506972560964878288625045457576007168893434587492892373016743739714521968994300'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33509a1}[Curve in LMFDB]+  '33609':+    11137, -1174049:+      number: '1.323855053459559861245458071960190988950554888912528348543289732810022588071850058311702388990078916'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33609a1}[Curve in LMFDB]+  '33709':+    2289, 46359:+      number: '0.951787640084913119908235799633262681020120452509776302712986828781037161174957962765162636581003503'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33709a1}[Curve in LMFDB]+  '33914':+    1321, -245845:+      number: '1.753020006794136717892865263884472698959174876178482352641967603943668416884985711792994914599139590'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/33914a1}[Curve in LMFDB]+  '34415':+    -431, -85753:+      number: '1.869733196319113781796106365263132776838652772618255706677406770307021487682742781124471069643381948'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/34415a1}[Curve in LMFDB]+  '34862':+    2721, -286065:+      number: '1.842652525062125643272418425797789774278359194902331264470491267732285060478926778242988344275099173'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/34862a1}[Curve in LMFDB]+  '35003':+    -848, 20456:+      number: '1.580582077477274410940161477363013622379682832005708279207563793163646882012075386994895564825568688'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35003a1}[Curve in LMFDB]+  '35083':+    208, -8344:+      number: '3.205391561209916228252777684178362604311754245324918240406953219936392033252507712518816879717249924'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35083c1}[Curve in LMFDB]+  '35234':+    8713, -712421:+      number: '0.919234874482560303418031282492257742841021776144187762902798136431979507361047115270185730018660642'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35234a1}[Curve in LMFDB]+  '35243':+    1057, -19729:+      number: '1.349452634851833356542715301593051052551024348182845401381993065444787596994499197450115141782662031'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/35243b1}[Curve in LMFDB]+  '35882':+    3081, -196965:+      number: 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'1.612189566744773864165807962222935075144020144623451684014531378161092915144764096957201190365787629'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/37544c1}[Curve in LMFDB]+  '38439':+    2736, -65880:+      number: '1.033088754317545077650856404045943668056904935144851855106234611285700275946747830138332039411835042'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/38439b1}[Curve in LMFDB]+  '38600':+    8400, -734400:+      number: '1.029306752721653364531769009339765350350082779368991368144836183757783124623865268841269614688958138'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/38600m1}[Curve in LMFDB]+  '38601':+    -144, -14040:+      number: '2.514078017493881485155408912446763869842916642372233168794008490114655877366719346278459042068107487'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/38601a1}[Curve in LMFDB]+  '39023':+    273, -9369:+      number: 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