The modular curve $X_{230f}$

Curve name $X_{230f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{102n}$
Meaning/Special name
Chosen covering $X_{230}$
Curves that $X_{230f}$ minimally covers
Curves that minimally cover $X_{230f}$
Curves that minimally cover $X_{230f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27648t^{16} + 110592t^{14} - 82944t^{12} - 27648t^{10} + 17280t^{8} - 6912t^{6} - 5184t^{4} + 1728t^{2} - 108\] \[B(t) = -1769472t^{24} + 10616832t^{22} - 18579456t^{20} + 6193152t^{18} + 6303744t^{16} - 3981312t^{14} - 774144t^{12} - 995328t^{10} + 393984t^{8} + 96768t^{6} - 72576t^{4} + 10368t^{2} - 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 257x - 3423$, with conductor $1344$
Generic density of odd order reductions $271/2688$

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