### Visualizing elements in the Shafarevich-Tate group.

Cremona, John E., Mazur, Barry (2000)

Experimental Mathematics

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Cremona, John E., Mazur, Barry (2000)

Experimental Mathematics

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Rubin, Karl, Silverberg, Alice (2000)

Experimental Mathematics

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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)

Acta Arithmetica

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Lisa Berger (2012)

Acta Arithmetica

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Andrej Dujella, Kálmán Győry, Ákos Pintér (2012)

Acta Arithmetica

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Gang Yu (2005)

Acta Arithmetica

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Ruthi Hortsch (2016)

Acta Arithmetica

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We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².

Joseph H. Silverman, Armand Brumer (1996)

Manuscripta mathematica

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Delaunay, Christophe (2001)

Experimental Mathematics

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Jörn Steuding, Annegret Weng (2005)

Acta Arithmetica

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Tom Fisher (2015)

Acta Arithmetica

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We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.

Alf Van Der Poorten (1980)

Mémoires de la Société Mathématique de France

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Kevin James, Gang Yu (2006)

Acta Arithmetica

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Noboru Aoki (2004)

Acta Arithmetica

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D.L. Ulmer (1990)

Inventiones mathematicae

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Jörn Steuding, Annegret Weng (2005)

Acta Arithmetica

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Jerzy Browkin, Daniel Davies

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We consider elliptic curves defined over ℚ. It is known that for a prime p > 3 quadratic twists permute the Kodaira classes, and curves belonging to a given class have the same conductor exponent. It is not the case for p = 2 and 3. We establish a refinement of the Kodaira classification, ensuring that the permutation property is recovered by {refined} classes in the cases p = 2 and 3. We also investigate the nonquadratic twists. In the last part of the paper we discuss the number...