# Odd degree isolated points on $X_{1}(N)$ with rational $j$-invariant

This website provides Magma scripts for computations done in relation to the paper mentioned above (authored by Abbey Bourdon, David Gill, Lori Watson, and Jeremy Rouse). You can find a link to the preprint here.

 Filename Description odd7.txt This script verifies that the mod 49 image of Galois for $E : y^{2} + xy = x^{3} - x^{2} - 107x - 379$ is the full preimage of the mod $7$ image. x02andX9C.txt This script computes the rational points on the fiber product $X_{0}(2) \times_{X_{0}(1)} X_{9C^{0}-9a}$. x02andX27A.txt This script computes the rational points on the fiber product $X_{0}(2) \times_{X_{0}(1)} X_{27A^{0}-27a}$. X9B0squaredisc.txt This script computes the rational points on the modular curve parametrizing elliptic curves whose discriminant is a square that also have a cyclic $9$-isogeny. X3D0squaredisc.txt This script computes the rational points on the modular curve parametrizing elliptic curves whose discriminant is a square and that also have two independent $3$-isogenies. X9J09asquaredisc.txt This script computes the rational points on the modular curve parametrizing elliptic curves whose discriminant is a square and that have mod $9$ image contained in $9J^{0}-9a$. S3entanglement.txt This script determines the possible mod $54$ images of Galois for an elliptic curve that gives rise to an odd degree point of degree $\leq 27$ on $X_{1}(54)$. deg3iso.txt Given two degree $3$ extension $K_{1}$ and $K_{2}$ of $\mathbb{Q}(x)$, for which values of $x$ are the specializations of $K_{1}$ and $K_{2}$ isomorphic? Instances are parametrized by rational roots of a degree $6$ polynomial in $\mathbb{Q}(x)[t]$ which this script computes. genus4.txt This script computes applies the previous one to compute the equation of the modular curve $X_{K}$ of genus $4$ and find all of its rational points. X128.txt This script verifies that the degree $9$ point above $j = 3^{3} \cdot 13/2^{2}$ on $X_{1}(28)$ is isolated. X127.txt This script verifies that the degree $9$ point above $j = -2^{15} \cdot 3 \cdot 5^{3}$ on $X_{1}(27)$ is not isolated.