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ID: INPUT{id.yaml}

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
  variable:
    display: 
    title: variable
    type: Symbolic
    show-in-parameter-list: no
   
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}.
    
#  comment-missing-curves: >
#    Two curves from CITE{BenYaz12} could not be preproduced at the moment,
#    see generate.sage.
    
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{Nit98}
  - CITE{BenYaz12}
  - CITE{LMFDB}
  - CITE{M-ECDB}

Display properties:
  number-header: value

Numbers: INPUT{numbers.yaml}

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

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Numbers

$\sigma'$

value

INPUT{numbers.yaml} (not shown in preview)

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

[4]

LMFDB: Elliptic curves over Q

[5]

Matschke - Tables of elliptic curves

[6]

Wikipedia: Szpiro's conjecture

Data properties

Entries are of type: real number

Table is complete: unknown (presumably not)

Sources of data: [3], [1], [4], [5]