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- Title
Explicit classification of isogeny graphs of rational elliptic curves.
- Authors
Barrios, Alexander J.
- Abstract
Let n > 1 be an integer such that X 0 (n) has genus 0 , and let K be a field of characteristic 0 or relatively prime to 6 n. In this paper, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a nontrivial isogeny over ℚ. We achieve this by introducing 5 6 parameterized families of elliptic curves n , i (t , d) defined over K (t , d) , which have the following two properties for a fixed n : the elliptic curves n , i (t , d) are isogenous over K (t , d) , and there are integers k 1 and k 2 such that the j -invariants of n , k 1 (t , d) and n , k 2 (t , d) are given by the Fricke parameterizations. As a consequence, we show that if E is an elliptic curve over a number field K with isogeny class degree divisible by n ∈ { 4 , 6 , 9 } , then there is a quadratic twist of E that is semistable at all primes of K such that ∤ n.
- Subjects
ELLIPTIC curves; SEMILINEAR elliptic equations; QUADRATIC forms; PARAMETERIZATION; INTEGERS
- Publication
International Journal of Number Theory, 2023, Vol 19, Issue 4, p913
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S179304212350046X