Computing modular degrees using L-functions
Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 673-682.

Nous donnons un algorithme pour calculer le degré modulaire d’une courbe elliptique définie sur ℚ. Notre méthode est basée sur le calcul de la valeur spéciale en s=2 du carré symétrique de la fonction L attachée à la courbe elliptique. Cette méthode est assez efficace et facile à implémenter.

We give an algorithm to compute the modular degree of an elliptic curve defined over ℚ. Our method is based on the computation of the special value at s=2 of the symmetric square of the L-function attached to the elliptic curve. This method is quite efficient and easy to implement.

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     title = {Computing modular degrees using $L$-functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {673--682},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
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     url = {http://archive.numdam.org/item/JTNB_2003__15_3_673_0/}
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Delaunay, Christophe. Computing modular degrees using $L$-functions. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 673-682. http://archive.numdam.org/item/JTNB_2003__15_3_673_0/

[1] A. Atkin, J. Lehner, Hecke operators on Γ0(m). Math. Ann. 185 (1970), 134-160. | EuDML | MR | Zbl

[2] A. Atkin, W. Li, Twists of newforms and pseudo-eigenvalues of W-operators. Invent. Math. 48 (1978), 221-243. | EuDML | MR | Zbl

[3] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 no. 4 (2001), 843-939. (electronic). | MR | Zbl

[4] J. Coates, C.-G. Schmidt, Iwasawa theory for the symmetric square of an elliptic curve. J. reine angew. Math. 375 (1987), 104-156. | EuDML | MR | Zbl

[5] H. Cohen, Advanced topics in computational algebraic number theory. Graduate Texts in Mathematics, 193, Springer-Verlag, New-York, 2000. | MR | Zbl

[6] J. Cremona, Algorithms for modular elliptic curves. Second edition, Cambridge University Press, 1997. | MR | Zbl

[7] J. Cremona, Computing the degree of the modular parametrization of a modular elliptic curve. Math. Comp. 64 (1995), 1235-1250. | MR | Zbl

[8] D. Goldfeld, J. Hoffstein, D. Lieman, An effective zero-free region. Ann. of Math. (2) 140 no. 1 (1994), 177-181. | MR | Zbl

[9] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, pari-gp, available by anonymous ftp.

[10] G. Shimura, The special values of the zeta functions associated with cusp forms. Com. Pure Appl. Math. 29 (1976), 783-804. | MR | Zbl

[11] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 no. 3 (1995), 553-572. | MR | Zbl

[12] E. Tollis, Zeros of Dedekind zeta functions in the critical strip. Math. Comp. 66 (1997), 1295-1321. | MR | Zbl

[13] M. Watkins, Computing the modular degree of an elliptic curve. Experimental Maths 11 no. 4 (2003), 487-502. | MR | Zbl

[14] A. Wiles, Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 no. 3 (1995), 443-551. | MR | Zbl

[15] D. Zagier, Modular parametrizations of elliptic curves. Canad. Math. Bull. 28 no. 3 (1985), 372-384. | MR | Zbl