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On the Computation of Class Polynomials with "Thetanullwerte" and Its Applications to the Unit Group Computation Leprévost, Franck ; ; in Experimental Mathematics (2011), 20(3), 271-281 Detailed reference viewed: 121 (0 UL)Jacobians of genus 2 curves with a rational point of order 11 Leprévost, Franck ; Bernard, Nicolas ; in Experimental Mathematics (2009), 18(1), 65-70 On the one hand, it is well-known that Jacobians of (hyper)elliptic curves defined over $\Q$ having a rational point of order $l$ can be used in many applications, for instance in the construction of ... [more ▼] On the one hand, it is well-known that Jacobians of (hyper)elliptic curves defined over $\Q$ having a rational point of order $l$ can be used in many applications, for instance in the construction of class groups of quadratic fields with a non-trivial $l$-rank. On the other hand, it is also well-known that $11$ is the least prime number which is not the order of a rational point of an elliptic curve defined over $\Q$. It is therefore interesting to look for curves of higher genus, whose Jacobians have a rational point of order $11$. This problem has already been addressed, and Flynn found such a family $\Fl_t$ of genus $2$ curves. Now, it turns out, that the Jacobian $J_0(23)$ of the modular genus $2$ curve $X_0(23)$ has the required property, but does not belong to $\Fl_t$. The study of $X_0(23)$ leads to a method to partially solving the considered problem. Our approach allows us to recover $X_0(23)$, and to construct another $18$ distinct explicit curves of genus $2$ defined over $\Q$ and whose Jacobians have a rational point of order $11$. Of these $19$ curves, $10$ do not have any rational Weierstrass point, and $9$ have a rational Weierstrass point. None of these curves are $\Qb$-isomorphic to each other, nor $\Qb$-isomorphic to an element of Flynn's family $\Fl_t$. Finally, the Jacobians of these new curves are absolutely simple. [less ▲] Detailed reference viewed: 203 (13 UL)On the failure of the Gorenstein property for Hecke algebras of prime weight ; Wiese, Gabor in Experimental Mathematics (2008), 17(1), 37--52 In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that ... [more ▼] In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular symbols algorithm over finite fields and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations. The Magma package, instructions for its use, generated tables and the complete data are available as supplemental material. [less ▲] Detailed reference viewed: 133 (1 UL)A database of invariant rings ; ; et al in Experimental Mathematics (2001), 10(4), 537--542 We announce the creation of a database of invariant rings. This database contains a large number of invariant rings of finite groups, mostly in the modular case. It gives information on generators and ... [more ▼] We announce the creation of a database of invariant rings. This database contains a large number of invariant rings of finite groups, mostly in the modular case. It gives information on generators and structural properties of the invariant rings. The main purpose is to provide a tool for researchers in invariant theory. [less ▲] Detailed reference viewed: 150 (1 UL)Finding the Eigenvalue in Elkies' Algorithm Müller, Volker ; in Experimental Mathematics (2001), 10(2), 275-285 One important part of Elkies' algorithm for computing the group order of an elliptic curve is the search for an eigenvalue of the Frobenius endomorphism. In this paper we compare two well known algorithms ... [more ▼] One important part of Elkies' algorithm for computing the group order of an elliptic curve is the search for an eigenvalue of the Frobenius endomorphism. In this paper we compare two well known algorithms with two new ideas based on the Babystep Giantstep method. Moreover we show how resultants can be used to speed up this search. Finally we present a fast probabilistic algorithm for checking whether a given rational function is congruent to an entry in a table of rational functions modulo some fixed polynomial. [less ▲] Detailed reference viewed: 132 (3 UL) |
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