Elliptic curves over $\mathbb{Q}$ with large Szpiro ratios
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Numbers
$\sigma'$
value
9.0199
$\sigma$:
9.019964068365010421032404283054510876688775648109581744943873471338203667169402662052010200968394701
9.0199
$N$:
12735814
9.0199
$c_4$:
20359284608016161208865
9.0199
$c_6$:
5436524013766214643338936120213327
8.9037
$\sigma$:
8.903700224703577364058979912566076297017598117884589976644796089959288132044681509092027329227305590
8.9037
$N$:
1290
8.9037
$c_4$:
-5771037719
8.9037
$c_6$:
2895537481474571
8.8431
$\sigma$:
8.843128226073368911990416647066625100656194890344662482250967294084911902806944845029710783781675002
8.8431
$N$:
9510
8.8431
$c_4$:
-1275806996039
8.8431
$c_6$:
16099757136577294291
8.8119
$\sigma$:
8.811943571940474883350816992201139371583018317306267688216100842830069019212129894458017437702422791
8.8119
$N$:
2526810
8.8119
$c_4$:
-16771680533237163431
8.8119
$c_6$:
669970461396005289047247621283
8.8015
$\sigma$:
8.801596471642693645149329698370532759326204415663971781853239507314244165513983396439557927816132634
8.8015
$N$:
9690
8.8015
$c_4$:
11840098430761
8.8015
$c_6$:
-43249124259565972309
8.7923
$\sigma$:
8.792374064160902757693706051988218987662136662032246275518000607316018687041196240878053198570425168
8.7923
$N$:
3990
8.7923
$c_4$:
9137595601
8.7923
$c_6$:
-281400139351426969
8.7826
$\sigma$:
8.782667844265430710013968626748761425035499984910681461137386071258495620282889428079039964041511676
8.7826
$N$:
32658
8.7826
$c_4$:
-10317044106455
8.7826
$c_6$:
2761698048773188730923
8.7573
$\sigma$:
8.757316145571119834804876236152506665462655784955531596970693059026377764138939092287578981318982755
8.7573
$N$:
858
8.7573
$c_4$:
-784948271
8.7573
$c_6$:
289910339991719
8.7210
$\sigma$:
8.721074059168133340936757361466370960978434922861075486193092786208677454390499941674063872693702937
8.7210
$N$:
843378
8.7210
$c_4$:
2035851168873923257
8.7210
$c_6$:
377954041919536430003420243
8.6989
$\sigma$:
8.698941971725240173631778688442416080239259633572821991336286429935397847350055952280280814900848931
8.6989
$N$:
89150698
8.6989
$c_4$:
997604945792791899234385
8.6989
$c_6$:
-1864727736721811622665255089233171161
8.6889
$\sigma$:
8.688967708221039435993719761328437032964051698684299815457621639053641602376432035943285027623161070
8.6889
$N$:
167490523410
8.6889
$c_4$:
14805784447948174261527465478951849
8.6889
$c_6$:
1785420779423319383954904305376561054007500373097707
8.6622
$\sigma$:
8.662217659460579400406354698354791505301849873246202509820380476339490716481833389062595352687089325
8.6622
$N$:
27107333238
8.6622
$c_4$:
113156356847769719661470180249929
8.6622
$c_6$:
1202002355169597637982306839532964842326661545915
8.6224
$\sigma$:
8.622430745489334182572517897465580672408913690756701899990543086365819049234760037571383103531735567
8.6224
$N$:
12735814
8.6224
$c_4$:
-2140997568706379744735
8.6224
$c_6$:
-147713378172370337753437116596401
8.6169
$\sigma$:
8.616929364024027180182008671129062056085934001924650016905964708583807301266344789548668438535005249
8.6169
$N$:
7580430
8.6169
$c_4$:
-150945124799134470879
8.6169
$c_6$:
-18089202457692142804275685774641
8.6106
$\sigma$:
8.610658659833089795143760884422276664687814165753910063742300402842054338708947580337800892401883427
8.6106
$N$:
165565582
8.6106
$c_4$:
3440719098754731244298185
8.6106
$c_6$:
11944043258244373571415642656108679419
8.5965
$\sigma$:
8.596580111291871506118386113177072037194578394082736912873707466794283349695925950455031484677567119
8.5965
$N$:
610537970
8.5965
$c_4$:
-108418460486773799492473871
8.5965
$c_6$:
-2131107242455933222484131795495627234633
8.5793
$\sigma$:
8.579323111854816123714544057470136263480704019285615840709961461612982451774007678446375836272753003
8.5793
$N$:
502471570230
8.5793
$c_4$:
133252060031533568353747189310566641
8.5793
$c_6$:
-48206361044429623366782416245167148458202510073638089
8.5593
$\sigma$:
8.559337741701674815983471619142866438357536315476710881960911809624589851057463556151236084005184798
8.5593
$N$:
241980466
8.5593
$c_4$:
7349701743493834196400265
8.5593
$c_6$:
-37289118210422466238661762848543209893
8.5457
$\sigma$:
8.545794523965473753175540158364337519499011845820688505771898963835238175190966127885519207053388491
8.5457
$N$:
81321999714
8.5457
$c_4$:
1018407211629927476953231622249361
8.5457
$c_6$:
-32454063589579136225522284667390050742819861739705
8.5386
$\sigma$:
8.538655717570655377253103323032695689944292017300518640422879265889333189667318951212168681142616567
8.5386
$N$:
28530
8.5386
$c_4$:
-11482262964351
8.5386
$c_6$:
-434693442687586945857
8.5372
$\sigma$:
8.537292953648903462341546456894528434656393877264221958072627105518564463887901029803181480170202522
8.5372
$N$:
361085848422
8.5372
$c_4$:
56134630838528340067322906196532873
8.5372
$c_6$:
13330236926194309924409343234354824150830582812943451
8.5351
$\sigma$:
8.535177758682160363405504201291652278231750576224431812488614782055945962499110980755714888286883560
8.5351
$N$:
12634050
8.5351
$c_4$:
-419292013330929085775
8.5351
$c_6$:
83746307674500661130905952660375
8.5318
$\sigma$:
8.531805122803815819066506231454608934204317243969545788225792828713221510975735645836534364502770260
8.5318
$N$:
573247290
8.5318
$c_4$:
71427406483335635565626809
8.5318
$c_6$:
-1131661295434174666745837205186845899373
8.5313
$\sigma$:
8.531330029976891656035100130462004605868995736273648241254237746048008709454329091385871863176409983
8.5313
$N$:
837452617050
8.5313
$c_4$:
370144611198704356538186636973796225
8.5313
$c_6$:
223177597427914922994363038172070131750937546637213375
8.5253
$\sigma$:
8.525311491260141405874174915481630261757835509685810454763661282141218193779898487700766103904041528
8.5253
$N$:
128310
8.5253
$c_4$:
4257946518705889
8.5253
$c_6$:
-126309683930544316277713
8.5175
$\sigma$:
8.517543951790708020085423173320667526219883246875950465389452812539310338686856798577821036587558182
8.5175
$N$:
3870
8.5175
$c_4$:
-51939339471
8.5175
$c_6$:
-78179511999813417
8.5166
$\sigma$:
8.516660933972458033544544784211473306412027390136766278999841315323825763194689400715482956191953390
8.5166
$N$:
97974
8.5166
$c_4$:
-92853396958095
8.5166
$c_6$:
-74565847316876095734921
8.5021
$\sigma$:
8.502119002052766164860204713612952235507200536926261201358163750379223673922152740260598809224419195
8.5021
$N$:
29070
8.5021
$c_4$:
106560885876849
8.5021
$c_6$:
1167726355008281252343
8.5012
$\sigma$:
8.501277293801510842261925078527485384697982537819576824173871417260444747556455629227571924647811386
8.5012
$N$:
532837573905
8.5012
$c_4$:
46875947016507288758867196829681849
8.5012
$c_6$:
-10558722942217181837747422248922106816631461912095757
8.5006
$\sigma$:
8.500681626687463855826261810935702356649515017265623690267129476308170194458793093372284289793642705
8.5006
$N$:
1172433663870
8.5006
$c_4$:
725483437949460538814845808468640601
8.5006
$c_6$:
-612399327342198548696532176744160441524572627972513501
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
$\sigma'$
—   Szpiro ratio (rounded)
(1)
The Szpiro conjecture [6] 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 [2].
References
[1]
Michael A. Bennett, Soroosh Yazdani, "A Local Version of Szpiro’s Conjecture", Experiment. Math. 21:2, 103-116 (2012). https://sf-lib-app-008.serverfarm.cornell.edu/download/pdf_1/euclid.em/1338430824
[2]
D. W. Masser. "Note on a conjecture of Szpiro.", In Les pinceaux de courbes elliptiques, Semin., Paris/Fr. 1988, Asterisque 183, 19-23 (1990).
[3]
A. Nitaj, "Détermination de courbes elliptiques pour la conjecture de Szpiro", Acta Arith. 85, 351–376 (1998).