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 modular forms of weight $$2$$, trivial character, and level $$N$$ equal to the conductor of the curve. The reliability of the completeness of the data thus relies on the rigour and correct implementation of the modular symbol algorithms used to compute them, together with the correctness of the algorithms used to compute complete isogeny classes from any one elliptic curve.

Individual curve data

Equation

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.

Conductor, local data and basic invariants

These are rigorously computed.

Mordell-Weil group and generators

The analytic rank $$r_{an}$$ is computed using modular symbols and is rigorous for $$r_{an}\le3$$. 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 height of generators of infinite order are given approximately; currently we do not guarantee the precision. Similarly for the regulator and the special L-value.

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 p Galois representations, and theimages of the 2-adic Galois representations, were computed rigorously.

Iwasawa invariants

These were computed rigorously.

Torsion growth

The torsion growth data was computed rigorously.