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Elliptic curves over $\mathbb{Q}$ with large Szpiro ratios
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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 $\sigma > 8.5$.
—   Szpiro ratio (rounded)
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].
Michael A. Bennett, Soroosh Yazdani, "A Local Version of Szpiro’s Conjecture", Experiment. Math. 21:2, 103-116 (2012).
D. W. Masser. "Note on a conjecture of Szpiro.", In Les pinceaux de courbes elliptiques, Semin., Paris/Fr. 1988, Asterisque 183, 19-23 (1990).
A. Nitaj, "Détermination de courbes elliptiques pour la conjecture de Szpiro", Acta Arith. 85, 351–376 (1998).
Data properties
Entries are of type: real number
Table is complete: unknown (presumably not)
Sources of data: [3], [1], [4], [5]