History of Elliptic curves over $\mathbb{Q}$ with large Szpiro ratios

+Title: Elliptic curves over $\mathbb{Q}$ with large Szpiro ratios+Definition: The Szpiro ratio of an elliptic curve over $\mathbb{Q}$ is  defined as+  $\sigma = \frac{\log |\Delta_E|}{\log N}$,  where $\Delta_E$ is the minimal discriminant+  of $E$ and  $N$ its conductor. This table lists all known Szpiro ratios $\sigma+  > 8.5$.+Parameters:+  sigma:+    display: $\sigma'$+    title: Szpiro ratio (rounded)+    type: R+Comments:+  comment-Szpiro-conjecture: The Szpiro conjecture CITE{Wiki} states that $\limsup+    \sigma = 6$, that is, for every $\varepsilon > 0$ there are only finitely many  $\mathbb{Q}$-isomorphism+    classes of elliptic curves $E/\mathbb{Q}$ with Szpiro ratio $\sigma > 6 + \varepsilon$.+    The lower bound $\limsup \sigma \geq 6$ was proved by Masser CITE{Mas88}.+Formulas: {}+Programs: {}+References:+  BenYaz12:+    bib: Michael A. Bennett, Soroosh Yazdani,  "A Local Version of Szpiro’s Conjecture",+      Experiment. Math. 21:2, 103-116 (2012).+    url: https://sf-lib-app-008.serverfarm.cornell.edu/download/pdf_1/euclid.em/1338430824+  Mas88:+    bib: D. W. Masser. "Note on a conjecture of Szpiro.", In Les pinceaux de courbes+      elliptiques, Semin., Paris/Fr. 1988,  Asterisque 183, 19-23 (1990).+  Nit98:+    bib: A. Nitaj,  "Détermination de courbes elliptiques pour la conjecture de Szpiro",+      Acta Arith. 85, 351–376 (1998).+Links:+  LMFDB:+    title: 'LMFDB: Elliptic curves over Q'+    url: https://www.lmfdb.org/EllipticCurve/Q/+  M-ECDB:+    title: Matschke - Tables of elliptic curves+    url: https://github.com/bmatschke/s-unit-equations/tree/master/elliptic-curve-tables/*+  Wiki:+    title: 'Wikipedia: Szpiro''s conjecture'+    url: https://en.wikipedia.org/wiki/Szpiro%27s_conjecture+Similar tables: ''+Keywords: ''+Tags:+- number theory+- elliptic curves+- abc conjecture+Data properties:+  type: R+  complete: unknown (presumably not)+  sources:+  - CITE{BenYaz12}+  - CITE{LMFDB}+  - CITE{M-ECDB}+Display properties:+  number-header: value+Numbers:+  '9.0199':+    sigma:+      param-latex: $\sigma$+      number: '9.019964068365010421032404283054510876688775648109581744943873471338203667169402662052010200968394701'+    N:+      param-latex: $N$+      number: '12735814'+    c4:+      param-latex: $c_4$+      number: '20359284608016161208865'+    c6:+      param-latex: $c_6$+      number: '5436524013766214643338936120213327'+  '8.8119':+    sigma:+      param-latex: $\sigma$+      number: '8.811943571940474883350816992201139371583018317306267688216100842830069019212129894458017437702422791'+    N:+      param-latex: $N$+      number: '2526810'+    c4:+      param-latex: $c_4$+      number: '-16771680533237163431'+    c6:+      param-latex: $c_6$+      number: '669970461396005289047247621283'+  '8.8015':+    sigma:+      param-latex: $\sigma$+      number: '8.801596471642693645149329698370532759326204415663971781853239507314244165513983396439557927816132634'+    N:+      param-latex: $N$+      number: '9690'+    c4:+      param-latex: $c_4$+      number: '11840098430761'+    c6:+      param-latex: $c_6$+      number: '-43249124259565972309'+  '8.7573':+    sigma:+      param-latex: $\sigma$+      number: '8.757316145571119834804876236152506665462655784955531596970693059026377764138939092287578981318982755'+    N:+      param-latex: $N$+      number: '858'+    c4:+      param-latex: $c_4$+      number: '-784948271'+    c6:+      param-latex: $c_6$+      number: '289910339991719'+  '8.7210':+    sigma:+      param-latex: $\sigma$+      number: '8.721074059168133340936757361466370960978434922861075486193092786208677454390499941674063872693702937'+    N:+      param-latex: $N$+      number: '843378'+    c4:+      param-latex: $c_4$+      number: '2035851168873923257'+    c6:+      param-latex: $c_6$+      number: '377954041919536430003420243'+  '8.6989':+    sigma:+      param-latex: $\sigma$+      number: '8.698941971725240173631778688442416080239259633572821991336286429935397847350055952280280814900848931'+    N:+      param-latex: $N$+      number: '89150698'+    c4:+      param-latex: $c_4$+      number: '997604945792791899234385'+    c6:+      param-latex: $c_6$+      number: '-1864727736721811622665255089233171161'+  '8.6889':+    sigma:+      param-latex: $\sigma$+      number: '8.688967708221039435993719761328437032964051698684299815457621639053641602376432035943285027623161070'+    N:+      param-latex: $N$+      number: '167490523410'+    c4:+      param-latex: $c_4$+      number: '14805784447948174261527465478951849'+    c6:+      param-latex: $c_6$+      number: '1785420779423319383954904305376561054007500373097707'+  '8.6622':+    sigma:+      param-latex: $\sigma$+      number: '8.662217659460579400406354698354791505301849873246202509820380476339490716481833389062595352687089325'+    N:+      param-latex: $N$+      number: '27107333238'+    c4:+      param-latex: $c_4$+      number: '113156356847769719661470180249929'+    c6:+      param-latex: $c_6$+      number: '1202002355169597637982306839532964842326661545915'+  '8.6224':+    sigma:+      param-latex: $\sigma$+      number: '8.622430745489334182572517897465580672408913690756701899990543086365819049234760037571383103531735567'+    N:+      param-latex: $N$+      number: '12735814'+    c4:+      param-latex: $c_4$+      number: '-2140997568706379744735'+    c6:+      param-latex: $c_6$+      number: '-147713378172370337753437116596401'+  '8.6169':+    sigma:+      param-latex: $\sigma$+      number: '8.616929364024027180182008671129062056085934001924650016905964708583807301266344789548668438535005249'+    N:+      param-latex: $N$+      number: '7580430'+    c4:+      param-latex: $c_4$+      number: '-150945124799134470879'+    c6:+      param-latex: $c_6$+      number: '-18089202457692142804275685774641'+  '8.6106':+    sigma:+      param-latex: $\sigma$+      number: '8.610658659833089795143760884422276664687814165753910063742300402842054338708947580337800892401883427'+    N:+      param-latex: $N$+      number: '165565582'+    c4:+      param-latex: $c_4$+      number: '3440719098754731244298185'+    c6:+      param-latex: $c_6$+      number: '11944043258244373571415642656108679419'+  '8.5965':+    sigma:+      param-latex: $\sigma$+      number: '8.596580111291871506118386113177072037194578394082736912873707466794283349695925950455031484677567119'+    N:+      param-latex: $N$+      number: '610537970'+    c4:+      param-latex: $c_4$+      number: '-108418460486773799492473871'+    c6:+      param-latex: $c_6$+      number: '-2131107242455933222484131795495627234633'+  '8.5793':+    sigma:+      param-latex: $\sigma$+      number: '8.579323111854816123714544057470136263480704019285615840709961461612982451774007678446375836272753003'+    N:+      param-latex: $N$+      number: '502471570230'+    c4:+      param-latex: $c_4$+      number: '133252060031533568353747189310566641'+    c6:+      param-latex: $c_6$+      number: '-48206361044429623366782416245167148458202510073638089'+  '8.5593':+    sigma:+      param-latex: $\sigma$+      number: '8.559337741701674815983471619142866438357536315476710881960911809624589851057463556151236084005184798'+    N:+      param-latex: $N$+      number: '241980466'+    c4:+      param-latex: $c_4$+      number: '7349701743493834196400265'+    c6:+      param-latex: $c_6$+      number: '-37289118210422466238661762848543209893'+  '8.5457':+    sigma:+      param-latex: $\sigma$+      number: '8.545794523965473753175540158364337519499011845820688505771898963835238175190966127885519207053388491'+    N:+      param-latex: $N$+      number: '81321999714'+    c4:+      param-latex: $c_4$+      number: '1018407211629927476953231622249361'+    c6:+      param-latex: $c_6$+      number: '-32454063589579136225522284667390050742819861739705'+  '8.5372':+    sigma:+      param-latex: $\sigma$+      number: '8.537292953648903462341546456894528434656393877264221958072627105518564463887901029803181480170202522'+    N:+      param-latex: $N$+      number: '361085848422'+    c4:+      param-latex: $c_4$+      number: '56134630838528340067322906196532873'+    c6:+      param-latex: $c_6$+      number: '13330236926194309924409343234354824150830582812943451'+  '8.5351':+    sigma:+      param-latex: $\sigma$+      number: '8.535177758682160363405504201291652278231750576224431812488614782055945962499110980755714888286883560'+    N:+      param-latex: $N$+      number: '12634050'+    c4:+      param-latex: $c_4$+      number: '-419292013330929085775'+    c6:+      param-latex: $c_6$+      number: '83746307674500661130905952660375'+  '8.5318':+    sigma:+      param-latex: $\sigma$+      number: '8.531805122803815819066506231454608934204317243969545788225792828713221510975735645836534364502770260'+    N:+      param-latex: $N$+      number: '573247290'+    c4:+      param-latex: $c_4$+      number: '71427406483335635565626809'+    c6:+      param-latex: $c_6$+      number: '-1131661295434174666745837205186845899373'+  '8.5313':+    sigma:+      param-latex: $\sigma$+      number: '8.531330029976891656035100130462004605868995736273648241254237746048008709454329091385871863176409983'+    N:+      param-latex: $N$+      number: '837452617050'+    c4:+      param-latex: $c_4$+      number: '370144611198704356538186636973796225'+    c6:+      param-latex: $c_6$+      number: '223177597427914922994363038172070131750937546637213375'+  '8.5021':+    sigma:+      param-latex: $\sigma$+      number: '8.502119002052766164860204713612952235507200536926261201358163750379223673922152740260598809224419195'+    N:+      param-latex: $N$+      number: '29070'+    c4:+      param-latex: $c_4$+      number: '106560885876849'+    c6:+      param-latex: $c_6$+      number: '1167726355008281252343'+  '8.5012':+    sigma:+      param-latex: $\sigma$+      number: '8.501277293801510842261925078527485384697982537819576824173871417260444747556455629227571924647811386'+    N:+      param-latex: $N$+      number: '532837573905'+    c4:+      param-latex: $c_4$+      number: '46875947016507288758867196829681849'+    c6:+      param-latex: $c_6$+      number: '-10558722942217181837747422248922106816631461912095757'+  '8.5006':+    sigma:+      param-latex: $\sigma$+      number: '8.500681626687463855826261810935702356649515017265623690267129476308170194458793093372284289793642705'+    N:+      param-latex: $N$+      number: '1172433663870'+    c4:+      param-latex: $c_4$+      number: '725483437949460538814845808468640601'+    c6:+      param-latex: $c_6$+      number: '-612399327342198548696532176744160441524572627972513501'