## The modular curve $X_{295}$

Curve name $X_{295}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 5 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{12}$ $8$ $24$ $X_{93}$
Meaning/Special name
Chosen covering $X_{93}$
Curves that $X_{295}$ minimally covers $X_{93}$, $X_{104}$, $X_{167}$
Curves that minimally cover $X_{295}$ $X_{563}$, $X_{564}$, $X_{565}$, $X_{566}$
Curves that minimally cover $X_{295}$ and have infinitely many rational points.
Model $y^2 = x^3 + x^2 - 9x + 7$
Info about rational points $X_{295}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 6627447363x - 207668330824122$, with conductor $1993828200$
Generic density of odd order reductions $2193/7168$