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Completeness of the collection

All elliptic curves defined over \(\mathbb{Q}\) are known to be modular, and hence arise (up to isogeny) via the Eichler-Shimura construction from classical modular forms of weight \(2\), trivial character, level \(N\) equal to the conductor of the curve, and having rational Fourier coefficients.

Individual curve data


The \(c_4\) and \(c_6\)-invariants of the optimal curve in each isogeny class were computed from numerical approximations obtained using modular symbols. See J. E. Cremona Algorithms for modular elliptic curves, 2nd edn., Cambridge 1997.

For additional justification that the equations obtained are rigorously correct, see J. E. Cremona, Appendix to a paper by Amod Agashe, Ken Ribet and William Stein: The Manin Constant, Pure and Applied Mathematics Quarterly, Vol. 2 no.2 (2006), pp. 617-636, supplemented for conductors over $130000$ by updated notes. Note that for conductors greater than $270000$ we have not always identified the optimal curve in each class rigorously, but expect that it is always the curve whose Cremona label number is 1.

Conductor , local data and basic invariants

These are rigorously computed.

Mordell-Weil group and generators, and BSD invariants

The analytic rank \(r_{an}\) is computed using modular symbols and is rigorous for \(r_{an}\le3\) and conductor \(N\le500000\). When \(r_{an}\le1\), it is a theorem that \(r_{an}\) equals the Mordell-Weil rank \(r\) of the curve. When \(r=1\) the generator is obtained from mwrank or using Heegner points. When \(r_{an}\ge2\), the Mordell-Weil rank and generators are obtained from mwrank. The torsion subgroup and generators are obtained using standard rigorous algorithms, based on Mazur's classification. When \(r_{an}\ge4\) we cannot compute the exact value, and when we claim that \(r_{an}=4\) we only know rigorously that \(r_{an}\in\{2,4\}\).

The heights of generators of infinite order are given approximately; currently we do not guarantee the precision. Similarly for the regulator, the real period and the special L-value. The analytic order of Ш was computed exactly for curves of rank $0$, where the quotient of the special L-value \(L(E,1)\) and the real period is a positive rational number, that was computed using modular symbols for curves with conductor \(N\le500000\) and using a rigorous algorithm of William Stein for larger conductors. For curves of positive rank the analytic order of Ш was computed approximately and rounded. Note that for rank $1$ curves, while it is possible in principal to compute the analytic order of Ш exactly, this has only been done for conductors less than $5000$. For curves of rank greater than 1, the quantity predicted by the BSD conjecture to be the order of Ш is not even known to be rational and can only be computed as a floating point approximation.

Integral points

These were computed rigorously, using independent implementations in Magma and SageMath which were compared as a consistency check.

Galois representations

The images of the mod-$\ell$ and $\ell$-adic Galois representations were computed rigorously. In particular, for the mod-$\ell$ representations a variant of Sutherland's algorithm [10.1017/fms.2015.33, arXiv:1504.07618] was used to rigorously compute the image of the mod-$\ell$ representation for all primes $\ell<1000$, and for elliptic curves over $\Q$ without complex multiplication Zywina's algorithm [arXiv:1508.07661] was used to verify the surjectivity of the mod $\ell$ representation for all primes $\ell \ge 1000$.

Iwasawa invariants

These were computed rigorously.

Torsion growth

The torsion growth data was computed rigorously.