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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)
Comments
(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).
Links
Data properties
Entries are of type: real number
Table is complete: unknown (presumably not)
Sources of data: [3], [1], [4], [5]
How well the digits are known: proven
How they were obtained: A fixed-precision value wrapped in an interval field, which records no error of its own. The working precision was half as many digits again as were written, and nothing checked that the margin sufficed.


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