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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)
(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).