History of Real periods of elliptic curves over $\mathbb{Q}$ of rank $2$

who	what
2026-08-13 22:07	bmatschke	how well the digits are known: assumed-bound (interval widened by a hand-chosen 4 ulps (blur_real_interval))	current
2026-08-09 09:15	flattening	entries rewritten as records with named parameters
2026-08-09 08:35	data-repository import	the current state of the data repository	reviewed
2021-05-08 23:03	bmatschke	from the data repository, 094738ca
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+Title: Real periods of elliptic curves over $\mathbb{Q}$ of rank $2$+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 $2$ 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..1000])\n                         if E.rank() ==\+      \ 2]"+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:+  '389':+    112, -856:+      number: '2.490212560855055075321357791942302460156058180339570040054972331731215476165102185993119256321325998'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/389a1}[Curve in LMFDB]+  '433':+    1, -865:+      number: '4.214710192998484367590655782878275529160375051005999082277668242586496123587134385822307428416825574'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/433a1}[Curve in LMFDB]+  '446':+    201, -2565:+      number: '2.414867176428977094737979611388156999986454895469380833137673887043361082906293486084590310212073923'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/446d1}[Curve in LMFDB]+  '563':+    721, -19385:+      number: '5.175209022515141907271535314239877251464358600610684258121499494473744016157076741855622366156858634'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/563a1}[Curve in LMFDB]+  '571':+    208, -3160:+      number: '5.095314619739130775182477273204610430607039496988259982361819720728241332880956052199107251497656487'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/571b1}[Curve in LMFDB]+  '643':+    193, -2881:+      number: '5.010313433142888912576313769140251226061947634543668367857675882832342237444940101107734248603839434'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/643a1}[Curve in LMFDB]+  '655':+    624, -15768:+      number: '4.487925804939284017569654367900108499649158177026887412252336953877777816186165658302809294821268841'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/655a1}[Curve in LMFDB]+  '664':+    336, -8640:+      number: '3.553627474682470749865951695397243506968892690233123445690524419676174366768917487912132257726417159'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/664a1}[Curve in LMFDB]+  '681':+    16, -1880:+      number: '3.770846168548564533409706831132855061765137317715721678069186029536009954479881282417612762107793674'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/681c1}[Curve in LMFDB]+  '707':+    592, -14104:+      number: '2.157259648094462244175812226334433906926474252080528622576825330873678996081582853637616074095641382'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/707a1}[Curve in LMFDB]+  '709':+    112, 424:+      number: '2.036664470021781648295044865870673994293140440352684399106230439176052424256977895319366635707324890'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/709a1}[Curve in LMFDB]+  '718':+    217, -541:+      number: '1.842389677901892516378196397754699875111168894929512218688205043502442470509332132723831947303901956'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/718b1}[Curve in LMFDB]+  '794':+    121, -2125:+      number: '4.342643373359145621151666617098962178096147650909496794568603864187498084736468577503878833512166112'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/794a1}[Curve in LMFDB]+  '817':+    -32, -5176:+      number: '3.056451464002115477168100079348260128618253721890044731913862924035110133472147731283256063846609880'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/817a1}[Curve in LMFDB]+  '916':+    192, -864:+      number: '1.951062787827085339010822633830829052938799041595387340064611158697277633589851351550966572192510479'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/916c1}[Curve in LMFDB]+  '944':+    912, -29376:+      number: '3.439691631192472544325521206268364843601872721561115708842756195862672488566712034488941276840233885'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/944e1}[Curve in LMFDB]+  '997':+    256, 3880:+      number: '1.576631347401765766383710183913607835699019692393415668783662071988439236599246786374653819158460099'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/997b1}[Curve in LMFDB]+    1168, -39896:+      number: '2.443448854153788226678335402437026518945420102576710055892704414316896506329810547914390877343512157'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/997c1}[Curve in LMFDB]+  '1001':+    9552, -943704:+      number: '2.281787535092866669084744898403659623361647587439956649808874886436121624806453584608714594787202775'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1001c1}[Curve in LMFDB]+  '1028':+    496, -10720:+      number: '2.181224977982202581601104841490343189958205596209659597809543064828797056852054832013744780270794904'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1028a1}[Curve in LMFDB]+  '1034':+    553, -13141:+      number: '4.665608258393240786254856635418656609404453595452162751268623500781576579289127278779024845280502652'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1034a1}[Curve in LMFDB]+  '1058':+    -23, -1909:+      number: '3.578388332687774502898094046623974785944935489466879197129618103097881006599122474005394414057813918'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1058c1}[Curve in LMFDB]+    5497, -407629:+      number: '3.578388332687774502898094046623974785944935489466879197129618103097881006599122474005394414057813918'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1058c2}[Curve in LMFDB]+  '1070':+    489, -11637:+      number: '3.961727130626772340749565372322641894569739832378587759119102732890731753946038430279307904232746105'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1070a1}[Curve in LMFDB]+  '1073':+    2176, -101240:+      number: 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'4.545434910812031519354841901611643384177319835657387817183131890448604634335897097946380870199823731'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1094a1}[Curve in LMFDB]+  '1102':+    1417, -63269:+      number: '2.739068212443759814964134797863072406190507064762581194388965205505528119489982247943788543854793581'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1102a1}[Curve in LMFDB]+  '1126':+    -87, -3861:+      number: '3.045635744168222149214782311825759624128690139568247286486222228464194549615619427786271603938744617'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1126a1}[Curve in LMFDB]+  '1132':+    256, -4960:+      number: '4.137098587630292641862860520649381215409551980219892842818558540085511765163470794410529204609814334'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1132a1}[Curve in LMFDB]+  '1137':+    97, -2609:+      number: '3.897103218265871511520313585024184589770786059893276712797651007924990919516084905585304087512730483'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1137a1}[Curve in LMFDB]+  '1141':+    1297, -83161:+      number: '2.307697978682582717243925896312906049780117439454936425777715248326488789182447609665195307257577653'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1141a1}[Curve in LMFDB]+  '1143':+    1872, -77976:+      number: '1.529960181647998990696949295611120331402834177258835050793929325716754544930158897319277818701111293'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1143c1}[Curve in LMFDB]+  '1147':+    448, -5336:+      number: '1.667050774832417421467437560590338073474085759768925141725890267087293120308444666895055200193995069'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1147a1}[Curve in LMFDB]+  '1171':+    129, 351:+      number: '1.990207204915774094029394287434086112713562541557162376963166745730471183750190220335246129845680545'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1171a1}[Curve in LMFDB]+  '1246':+    57, -10989:+      number: '2.815130249974162377659004976640637426423788140618068801667252498161143151680372301135709015941815628'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1246c1}[Curve in LMFDB]+  '1309':+    1072, -38744:+      number: '3.145535071333299272224750717980412681132708765735355595082361949838527636812444545999559975582435917'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1309b1}[Curve in LMFDB]+  '1324':+    -128, -2656:+      number: '3.004434943686204108496576716250759350991779167331206869799630627194381162325794627127238516492132065'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1324a1}[Curve in LMFDB]+  '1325':+    400, 2600:+      number: '1.486409035515803869345958385218630665449769772871068096757391610413561007303161640074615385526308666'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1325e1}[Curve in LMFDB]+  '1431':+    1377, 28431:+      number: '1.067675118414473185796001452238736706779841247586730625256685263232549896057060591379328044167767172'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1431a1}[Curve in LMFDB]+  '1436':+    592, -6976:+      number: '1.523369079146030476664913828740065217626698137584462702510482925951071852708651950427162124625641669'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1436a1}[Curve in LMFDB]+  '1443':+    433, -8585:+      number: '2.155041607178865512908403702356590495241584441724131166530873122935414303685583131532819622022651048'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1443c1}[Curve in LMFDB]+    -287, -34289:+      number: '2.155041607178865512908403702356590495241584441724131166530873122935414303685583131532819622022651048'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1443c2}[Curve in LMFDB]+  '1446':+    217, -5021:+      number: '3.776443259747556351870929274048126473834337200825257327104403449598003929099690474405833555705660753'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1446a1}[Curve in LMFDB]+  '1466':+    2001, -81945:+      number: '1.384007877114427541733141151905906517590203833916817459999715345234338477125519840035108399233991196'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1466b1}[Curve in LMFDB]+  '1477':+    289, -6481:+      number: '3.806688220734765507350811614128145714215092292471059973955537567723666924157072081946856538184689092'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1477a1}[Curve in LMFDB]+  '1480':+    1344, -44928:+      number: '1.521953753991208497595640476834329481680331273600510546056936099864657512097227674528618228719453285'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1480a1}[Curve in LMFDB]+  '1483':+    -80, -1432:+      number: '3.362058699602772257694482914662172739835628272006301389965021166582623185735009552814436887890090430'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1483a1}[Curve in LMFDB]+  '1525':+    385, -6625:+      number: '2.006782470011628766929762893201961420331523007928398011671355847392326606999148005364481588694801424'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1525c1}[Curve in LMFDB]+    1585, 56375:+      number: '1.003391235005814383464881446600980710165761503964199005835677923696163303499574002682240794347400712'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1525c2}[Curve in LMFDB]+  '1531':+    672, -17496:+      number: '4.791222836254008344837020364458411172334482067327142820516972743628963565313433849518046037379862752'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1531a1}[Curve in LMFDB]+  '1534':+    -231, -33021:+      number: '2.190784869367357304282870506775731812779331159032693280206027199061172508390016308403885486410547644'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1534b1}[Curve in LMFDB]+  '1570':+    169, -5653:+      number: '3.445066262192247925694964600505025193050662190601542810153692075431575973818178007860980684444440095'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1570b1}[Curve in LMFDB]+  '1576':+    448, 1664:+      number: '1.468290300928090536272176378841705500414067285667253825600677950004611373124780837997656826731258814'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1576a1}[Curve in LMFDB]+  '1591':+    3408, -477144:+      number: '1.656712603576273459941299231742292163276699326786795085883843003504770493717419871426121091112436528'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1591a1}[Curve in LMFDB]+  '1594':+    1281, -59265:+      number: '2.653439602869024603308493765778825285973273963732542003218882442178755708084882406609266608114708675'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1594a1}[Curve in LMFDB]+  '1608':+    1216, -45440:+      number: '3.176032605685443312453442624170450348734741065257746021072581299662281006702630292927020843155107690'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1608a1}[Curve in LMFDB]+  '1611':+    432, -17496:+      number: '2.949738718235335115771598313056562933439205636954110611170151162319565891654721628945963628171964179'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1611d1}[Curve in LMFDB]+  '1613':+    160, -1144:+      number: '2.157984673005884082462759950538954070237899328959136329967165755189790186114291643469640662857783334'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1613a1}[Curve in LMFDB]+  '1615':+    10321, -1045369:+      number: 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     comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1639b1}[Curve in LMFDB]+  '1641':+    208, -4184:+      number: '4.017299891189998766901676311183897189127826421963605055206574862834479541551874023182056528517453754'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1641b1}[Curve in LMFDB]+  '1642':+    73, -4805:+      number: '3.321937640640513931489244880090213728901848953812194066638697519661403587714060607604335008169097749'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1642a1}[Curve in LMFDB]+  '1653':+    1312, -149624:+      number: '1.961188976609796819657037006908649439091910560436308134039158503918874129611976142426963800287834879'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1653a1}[Curve in LMFDB]+  '1662':+    1321, -48725:+      number: '3.637937505351789301381752314042423959882674425255703336457078823461283339602829981207907887977040708'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1662a1}[Curve in LMFDB]+  '1664':+  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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1922b2}[Curve in LMFDB]+  '1933':+    -47, 1799:+      number: '2.275360631096268595431746578892396310906805067637344886096243209774146327334756519322731927110265226'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1933a1}[Curve in LMFDB]+  '1952':+    832, -31744:+      number: '2.918106024718541924866670665416027976694718056159012534721714900295010507119898568779412913967774318'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1952b1}[Curve in LMFDB]+  '1957':+    25081, -3916205:+      number: '0.876579933177686621701474438278374198643341899272014400740469226561134905205900365092886735872404563'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1957a1}[Curve in LMFDB]+    385, 7327:+      number: '1.421439385135460828613156687983523941935328545460979395966932235361661725196796772411174993478445569'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/1957b1}[Curve in LMFDB]+  '1964':+    768, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2458b1}[Curve in LMFDB]+    2737, -108649:+      number: '1.138753427587490475884670445095696999917495879162880958219478551888341646083176244961075312811874389'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2458d1}[Curve in LMFDB]+  '2465':+    2449, -121177:+      number: '2.217625812444869554336773973565406798667305939550720987281309061534934167355052285627781676396598780'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2465a1}[Curve in LMFDB]+    2209, -145873:+      number: '2.217625812444869554336773973565406798667305939550720987281309061534934167355052285627781676396598780'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2465a2}[Curve in LMFDB]+  '2466':+    1737, -85509:+      number: '2.611732449263189495174303278683082263768951924197434990803182340638286223614817066297294472114508405'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2466d1}[Curve in LMFDB]+  '2474':+    273, -17145:+   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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2735c1}[Curve in LMFDB]+  '2736':+    7056, -527040:+      number: '0.981047881676030025348176632628315241157055948142528052887237409241302397220196924006750369863222927'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2736h1}[Curve in LMFDB]+    -10224, -2704320:+      number: '0.981047881676030025348176632628315241157055948142528052887237409241302397220196924006750369863222927'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2736h2}[Curve in LMFDB]+  '2738':+    2257, -107929:+      number: '3.419592018837855130563526569631187777011489879240246613019075819220525066681115409356570066493147993'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2738d1}[Curve in LMFDB]+    -6623, -574129:+      number: '1.139864006279285043521175523210395925670496626413415537673025273073508355560371803118856688831049331'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/2738d2}[Curve in LMFDB]+  '2742':+    553, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3455c1}[Curve in LMFDB]+  '3456':+    4032, -262656:+      number: '2.611668596614117699948250675175269528308623007687939830820644923179095453235075979407944283012195959'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3456l1}[Curve in LMFDB]+  '3479':+    1281, -46305:+      number: '3.799872619666875671132800884476981944011674796922007904230171440759354770213524044612321231534907928'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3479e1}[Curve in LMFDB]+  '3483':+    513, 8991:+      number: '1.342330993935240632484784333756000114511770853484731508424302917035380204619963731377562653095019442'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3483f1}[Curve in LMFDB]+  '3489':+    256, -13400:+      number: '2.923618880238117891748473256776063610167813840022983290942517568930954737847252909873150958505384907'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3489a1}[Curve in LMFDB]+    2320, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3779a1}[Curve in LMFDB]+  '3782':+    6297, -501309:+      number: '2.804655435378882173771192961938102788330002035089381445794265645077662854267088333847978074077637589'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3782a1}[Curve in LMFDB]+  '3784':+    -9536, 470656:+      number: '0.911499625742482033223913081931078111872998282900507057327901467398358914451215765313640912450996870'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3784f1}[Curve in LMFDB]+  '3786':+    -167, -3869:+      number: '2.814579255858933594644137061106420423087800650443000256547982009240509513768599841583730774292351816'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3786a1}[Curve in LMFDB]+    9841, -1016585:+      number: '1.994868468929732669069482861210476200795636850751449572213804442849386300306114358083856733906812945'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/3786c1}[Curve in LMFDB]+  '3799':+    208, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4547a1}[Curve in LMFDB]+  '4554':+    -567, -2565:+      number: '2.044763713061443320103620454276239225354123109528040789725825768131196397475259707174452064140439234'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4554b1}[Curve in LMFDB]+    2313, -20709:+      number: '1.022381856530721660051810227138119612677061554764020394862912884065598198737629853587226032070219617'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4554b2}[Curve in LMFDB]+  '4564':+    -48, 22464:+      number: '1.427486172565741787579071717381226136658024941327415305436668397200232595382238494357173617799763108'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4564a1}[Curve in LMFDB]+  '4569':+    1393, -28009:+      number: '1.247145715958290743970547332792801400094074067079900526574943138044161651701900836835464687022851398'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4569b1}[Curve in LMFDB]+  '4572':+    1152, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4592c2}[Curve in LMFDB]+    30864, -5420736:+      number: '1.128326723451544977117089988360534125650246061294884646534739821180514487787106603676830042137659826'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4592g1}[Curve in LMFDB]+  '4593':+    -368, -12824:+      number: '2.308487094789999039652755774930432925561306865504408245037880362705550967250510956321969461010430527'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4593a1}[Curve in LMFDB]+  '4595':+    -119, -2501:+      number: '3.051241999999701874544380101279050473602661881997637515271347317006094949988135587062277085759739020'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4595a1}[Curve in LMFDB]+  '4600':+    -320, -35200:+      number: '2.133489578697484900774749792631496716724158502280083652697700005849419348476923055561589406441432582'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4600g1}[Curve in LMFDB]+  '4605':+    784, 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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4659b1}[Curve in LMFDB]+  '4662':+    3033, -370845:+      number: '1.742876757852746409450724873083354951746636951255408344300919691453883109628251407190613196074366341'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4662b1}[Curve in LMFDB]+    63513, -15992829:+      number: '0.871438378926373204725362436541677475873318475627704172150459845726941554814125703595306598037183170'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4662b2}[Curve in LMFDB]+  '4668':+    3712, -192896:+      number: '1.116544728451120058208831306038464472764981564920817784134296884649317135435643938493136777542190755'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4668a1}[Curve in LMFDB]+  '4669':+    21697, -3205889:+      number: '2.065282001126497823412982434578682877162398137054532078501471717403663711737949053587090766020467992'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4669b1}[Curve in LMFDB]+  '4675':+    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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4768a1}[Curve in LMFDB]+  '4776':+    3472, -204704:+      number: '3.698264926954557439955568811347711552060256403780401377096227704953682489355402668160857389090741918'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4776b1}[Curve in LMFDB]+  '4799':+    1489, -57529:+      number: '4.327613335857386116544808139865191201991250886893492532784177967950947365843222668202948926007483784'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4799c1}[Curve in LMFDB]+  '4806':+    8001, -599265:+      number: '0.911147536994300505496117299123985272828495489761812229017894098564522880389792873684234132518381003'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4806e1}[Curve in LMFDB]+  '4811':+    865, -28081:+      number: '3.319081697188342612555246574773897944396842052648920368022647634021403853051821161417973968438558881'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/4811a1}[Curve in LMFDB]+  '4836':+  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comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5472g2}[Curve in LMFDB]+  '5484':+    -128, -5984:+      number: '2.808689964382853859074050841506676749898357755201718122792164124024942203422774188970107069290750191'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5484a1}[Curve in LMFDB]+    21952, -3290464:+      number: '1.841250462605919877665181042741464661435862980915497647171645052488191693535353172740343675393674404'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5484d1}[Curve in LMFDB]+  '5497':+    6145, -481697:+      number: '1.983125971417951281302883954547005898508178965296940534121209003019172649382798932952572755432070573'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5497a1}[Curve in LMFDB]+  '5499':+    14832, -1729944:+      number: '0.900596941006445424722846351079818205318211300359584616831155288346034593694926423642557014649130139'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5499f1}[Curve in LMFDB]+  '5510':+    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    comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5727i1}[Curve in LMFDB]+  '5732':+    336, -25920:+      number: '2.576811844341915979498942628347886385839815542034658015725952772545404420489359883624617416224117244'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5732a1}[Curve in LMFDB]+  '5734':+    567609, 432179091:+      number: '0.228743051350119530302663181274994052145754660669842924911419734583852510903504928036849918207628540'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5734a1}[Curve in LMFDB]+  '5736':+    -368, -3104:+      number: '2.322429767160751676611762227603934651069642568984114941814502421005343897282952845766436241932843854'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5736f1}[Curve in LMFDB]+  '5741':+    249, -2349:+      number: '1.950641103918951703314032771517529877882953386768173807456196508921716825549444515857198141960054163'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5741a1}[Curve in LMFDB]+  '5754':+    10873, -1132109:+      number: '1.301447050208728788036416610805362860055194426663955333186268722082234616405876796313313650070351916'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5754a1}[Curve in LMFDB]+    7513, -1844765:+      number: '1.301447050208728788036416610805362860055194426663955333186268722082234616405876796313313650070351916'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5754a2}[Curve in LMFDB]+  '5760':+    -2016, -79488:+      number: '1.535696208536684140279885575587602254687686967768086122802018671611778735512484660321106963142574869'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5760r1}[Curve in LMFDB]+    10944, -732672:+      number: '0.767848104268342070139942787793801127343843483884043061401009335805889367756242330160553481571287434'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5760r2}[Curve in LMFDB]+  '5763':+    -19376, -1649944:+      number: '0.868537404320620611722967545277241446854349495518908640065095640273562605311198743259540144821695343'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5763d1}[Curve in LMFDB]+  '5764':+    768, -86400:+      number: '2.112738917373124356695822172119745046828242735412769855644950589555873120032840216818585221466142869'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5764a1}[Curve in LMFDB]+  '5772':+    103936, -33355360:+      number: '0.676959890193674739352953407848103343976252369455719889647770181841226064292832148974544580874388706'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5772d1}[Curve in LMFDB]+    162256, 8285120:+      number: '0.338479945096837369676476703924051671988126184727859944823885090920613032146416074487272290437194353'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5772d2}[Curve in LMFDB]+  '5781':+    4192, -264536:+      number: '1.298233332710074399597143732162623444485880375282458677205803846543158541107870271857727183924500424'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5781d1}[Curve in LMFDB]+    145264, 49048424:+      number: '0.324471191615925581289011046783248687842787461661360395334685167647630112669722814885614262213909711'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5781h1}[Curve in LMFDB]+  '5784':+    4048, -255680:+      number: '1.465609737198675318847414170647566158389322444394555186709807362420087394538444242251523535346306150'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5784a1}[Curve in LMFDB]+    6928, 155584:+      number: '0.732804868599337659423707085323783079194661222197277593354903681210043697269222121125761767673153075'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5784a2}[Curve in LMFDB]+  '5789':+    -272, -7064:+      number: '2.506701541812946535582515893228923313874904030164956973671725066818657415037412828992115234906773174'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5789c1}[Curve in LMFDB]+  '5793':+    577, -148609:+      number: '1.856142205931920167720321601202573405770786704982234074247756526470145480910279146071329291108469054'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5793f1}[Curve in LMFDB]+  '5794':+    66913, -17311121:+      number: '1.947092246688565060596551810185714028150560740464309062118426309944422707424990672549028378866763629'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5794a1}[Curve in LMFDB]+    -447, -15201:+      number: '2.211950204882080808321856230478285058698853559751097777365365964874928338287567846983461769974348961'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5794c1}[Curve in LMFDB]+  '5798':+    1160041, -1248269365:+      number: '0.418921591593435857918770638746866863147989840016666704670830014197301031997358677035952054710555818'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5798a1}[Curve in LMFDB]+    3865, -239741:+      number: '1.630473225917400829549321711496770614490146365244477286745724492730446907017828192498133013918613981'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5798b1}[Curve in LMFDB]+  '5799':+    481, 10063:+      number: '1.346010985225715593045908972019094571659022130629648903838503447978223292560177164711326200794073651'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5799a1}[Curve in LMFDB]+  '5805':+    18144, -2746872:+      number: '1.519746041954362621826136587297180897729045852112909568729385609010942592406748333701296321616151437'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5805e1}[Curve in LMFDB]+  '5808':+    44176, -9254080:+      number: '0.860114332153393711351078930930635128621633750203038766343701377846565570298700709933436072842070242'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5808k1}[Curve in LMFDB]+    65296, 503360:+      number: '0.430057166076696855675539465465317564310816875101519383171850688923282785149350354966718036421035121'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5808k2}[Curve in LMFDB]+  '5813':+    240, -1944:+      number: '1.927665626394307158642832412838995427206995572741155104613303867942491995102654458629274756844760701'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5813a1}[Curve in LMFDB]+  '5814':+    1449, -55701:+      number: '3.687932727951999271394222898418270893488259675171127839044970641463483766093672114062536803429619369'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5814b1}[Curve in LMFDB]+    -2583, 828171:+      number: '0.815621621210583188533768567389772832160708664809576729997124038776470310748399812920870576651360615'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5814k1}[Curve in LMFDB]+    23337, -22849749:+      number: '0.815621621210583188533768567389772832160708664809576729997124038776470310748399812920870576651360615'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5814k2}[Curve in LMFDB]+  '5826':+    35737, -6768829:+      number: '1.892745366426930832160865366413642265576654732095507187396234337748813161240646330788080600645960849'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5826b1}[Curve in LMFDB]+  '5830':+    33969, -6350697:+      number: '1.620256837985364818904573416884134854497961612853131280618305336317690169856290677895268729464551296'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5830f1}[Curve in LMFDB]+  '5834':+    7633, -338201:+      number: '0.808468814677728626496120003380813194172239834886637013278012544296152086417277646659499773675503165'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5834b1}[Curve in LMFDB]+  '5840':+    37776, -7286976:+      number: '0.836048966721895887590213818224435939438920280458785750359797706359379535802858437600744807214760399'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5840a1}[Curve in LMFDB]+    11536, 1009216:+      number: '0.614492720809794713639893914943383398974387096360006961433522311609346231644810683226681341791346992'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5840l1}[Curve in LMFDB]+    280336, -148336064:+      number: '0.614492720809794713639893914943383398974387096360006961433522311609346231644810683226681341791346992'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5840l2}[Curve in LMFDB]+  '5848':+    2464, -144224:+      number: '2.396372997811619229388496112562643724445792467186212898720933657621174614929781280414855386845302946'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5848c1}[Curve in LMFDB]+  '5854':+    83329, -24068161:+      number: '1.674651230154500765647144277079995083834604099181318474312570682182336629702192424137558684847397945'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5854b1}[Curve in LMFDB]+    -863, -2161:+      number: '1.820087640696160873645598031999880041944594715476305226192213607617018849015690721997990754850559766'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5854c1}[Curve in LMFDB]+  '5856':+    -3008, -88064:+      number: '1.379801331956909241010546242171267649721678851568495014000413520671079860039427837843319165697821299'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5856d1}[Curve in LMFDB]+  '5862':+    24817, -3845321:+      number: '0.866620745451262380821404916263172647372145113228118687017378764506419863979648549077120386839730800'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5862i1}[Curve in LMFDB]+  '5864':+    1168, -17216:+      number: '1.269680901541250460816011665255419929655161486211540592237959688667200937653753054622411367434751314'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5864a1}[Curve in LMFDB]+  '5871':+    58417, -14141321:+      number: '1.700396787752173351118136634972091128374650318149847068700794923342128054882848021409299452005835255'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5871b1}[Curve in LMFDB]+    41104, -8348696:+      number: '1.835266759907186140482669327764020056442069299691789048476539261316863930985275221953584614215793981'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5871c1}[Curve in LMFDB]+  '5886':+    3681, -195345:+      number: '1.138910724203390676604618168765376054281672784859059152807793689070832525513714740878410976823057687'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5886g1}[Curve in LMFDB]+  '5891':+    -1040, 16424:+      number: '1.590432553744099487209537100650774272700176543944629751897369176531270312414736708038966958463320067'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5891a1}[Curve in LMFDB]+  '5895':+    5409, -413073:+      number: '2.334305069899720323304610538692300844121770102194234808266370722667386501049269642545304357713801789'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5895a1}[Curve in LMFDB]+  '5904':+    4752, -333504:+      number: '2.593607846614939809462152344823743359784745013056314632424963203346666571251609240506379923040029047'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5904b1}[Curve in LMFDB]+    10512, -3547584:+      number: '1.153244530172663152805826449578304079824484208013782100962048578643573368345985096157223032979545661'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5904v1}[Curve in LMFDB]+    -93168, 87898176:+      number: '0.384414843390887717601942149859434693274828069337927366987349526214524456115328365385741010993181887'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5904v2}[Curve in LMFDB]+  '5906':+    1561, -61757:+      number: '4.255857229985353785863708136734562001988171958557409028963470400682470138135989811949495779994172966'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5906b1}[Curve in LMFDB]+    -303, -50841:+      number: '2.040314895187630332391194728953660312415358620473649361733614460787595515368493673903176270543495858'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5906c1}[Curve in LMFDB]+  '5907':+    -128, -18296:+      number: '2.437712238547994643147057014822650065377745330445828879813907195714549863947055937025571244205073576'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5907c1}[Curve in LMFDB]+  '5918':+    42153, -8654229:+      number: '1.191967334069726064349330595016782373318029304542155899288540035381050073247289050233590546432950583'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5918b1}[Curve in LMFDB]+    961, -15265:+      number: '1.359050068636479201100294187689259185205199833185131454452256360570232786616623067931198934170606628'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5918c1}[Curve in LMFDB]+  '5920':+    2944, 137728:+      number: '0.861470263467105192457574132997837366604312650619944207958378806380085212251082387513354987546923781'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5920i1}[Curve in LMFDB]+  '5922':+    11673, -1417149:+      number: '1.697308162986669893839913000550173758593547375757741285707048633340455912820959502306791322948724462'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5922e1}[Curve in LMFDB]+    193113, -84861405:+      number: '0.848654081493334946919956500275086879296773687878870642853524316670227956410479751153395661474362231'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5922e2}[Curve in LMFDB]+  '5928':+    273472, -143156096:+      number: '1.194151816445487242469400139099482390298035342023244274055583611310432980707749860643813904804236741'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5928b1}[Curve in LMFDB]+  '5930':+    5881, -462221:+      number: '2.385764765770366252931634750785054979835601266363757257651074602134073798932102958857251839680247809'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5930e1}[Curve in LMFDB]+    10369, -1397953:+      number: '1.552302381639532039725541731614650809582147660056039581973445617243811887984793171475144805486457851'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5930g1}[Curve in LMFDB]+  '5932':+    2688, -141696:+      number: '3.012175712300306088102661887266838320783840373568947514511925551159078682978227810194811991104002915'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5932a1}[Curve in LMFDB]+  '5934':+    132217, -52506829:+      number: '0.956658317599117817758536565003656709554287852886909933716255808725276305325012646466555203822851452'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5934d1}[Curve in LMFDB]+  '5954':+    9777, -964521:+      number: '1.291353309262047735680272604460331466406268009671017672805236198766907236707011792037753725852870483'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5954h1}[Curve in LMFDB]+  '5956':+    496, 8992:+      number: '1.349511825989391574050840781667428649742455651904619250606032670402735378240520216863352263250027959'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5956a1}[Curve in LMFDB]+  '5962':+    697, -2989:+      number: '1.374761705219300697906475888777705501343847936513153187670149025162527318732212328644959089297752483'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5962a1}[Curve in LMFDB]+  '5966':+    9433, -916381:+      number: '3.069354251608302139877761637052659517881007842379435730798596512189441694171883980256167675407200768'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5966b1}[Curve in LMFDB]+  '5970':+    4969, -444853:+      number: '1.910468312998990305742856465560073533377703756263201018832231369680324695330554831596498984033037522'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5970d1}[Curve in LMFDB]+  '5978':+    553, -4949:+      number: '1.514204657469690366477303146964405271504571560480601767168503622678321181011808383977819344853311089'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5978c1}[Curve in LMFDB]+  '5982':+    44569, -9409661:+      number: '2.276501118881161251265369736139385911119015222394132199517045983096608405444552643712843722985143346'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5982h1}[Curve in LMFDB]+    1297, -100601:+      number: '2.174225901077382999629751606557460891357142179271114497903738788813460716454713189684097080488345934'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5982l1}[Curve in LMFDB]+  '5988':+    7552, -663904:+      number: '2.405625354237064929393486586710001258596639075183595677013684067406662623682021453980754734785469660'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5988b1}[Curve in LMFDB]+    -27008, -3349216:+      number: '0.801875118079021643131162195570000419532213025061198559004561355802220874560673817993584911595156553'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5988b2}[Curve in LMFDB]+  '5992':+    11488, -1365344:+      number: '1.729266585323156477712096508843067526543300523470758666882283165659666672798610342998727427899052495'+      comment: HREF{https://www.lmfdb.org/EllipticCurve/Q/5992c1}[Curve in LMFDB]
History of Real periods of elliptic curves over $\mathbb{Q}$ of rank $2$

The first version, 2021-05-08 23:03
