## Calculating root numbers of elliptic curves over $$\mathbb{Q}$$.(English)Zbl 0805.14017

Let $$E$$ be an elliptic curve defined over $$\mathbb{Q}$$. For each prime number $$p$$ and the ‘prime at infinity’ $$p = \infty$$ there is associated to $$E$$ the local root number $$W_ p = W_ p(E)$$. These are intrinsic invariants of $$E$$. For all $$p$$, $$W_ p = \pm 1$$, and for all but finitely many $$p$$, $$W_ p=1$$. The “root number” is then defined to be $$W(E) = \prod_{p \leq \infty} W_ p (E)$$. This plays an important role in number theory, and because of this it is desirable to be able to write formulas for $$W$$ that can be immediately translated into computer code. The present paper builds on an earlier paper of D. E. Rohrlich [Compos. Math. 87, No. 2, 119-151 (1993; Zbl 0791.11026)]. This latter paper determines $$W_ p(E)$$ for all $$p \geq 5$$ and all $$E$$ over $$\mathbb{Q}$$, and for $$p=2, 3$$ and certain $$E$$. Many of these determinations are quite explicit, but some of the difficult cases require some nontrivial work to put the results in a form ready for computer implementation. The present paper carries out this work. For technical reasons, it is assumed throughout that the $$j$$-invariant of $$E$$ is not 0 or 1728.

### MSC:

 14H52 Elliptic curves 11G05 Elliptic curves over global fields 14Q05 Computational aspects of algebraic curves

Zbl 0791.11026

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### References:

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