The modular curve $X_{213j}$

Curve name $X_{213j}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{102i}$
Meaning/Special name
Chosen covering $X_{213}$
Curves that $X_{213j}$ minimally covers
Curves that minimally cover $X_{213j}$
Curves that minimally cover $X_{213j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{24} - 2160t^{22} - 15768t^{20} - 48816t^{18} - 48276t^{16} + 30240t^{14} + 59184t^{12} + 30240t^{10} - 48276t^{8} - 48816t^{6} - 15768t^{4} - 2160t^{2} - 108\] \[B(t) = 432t^{36} + 12960t^{34} + 159408t^{32} + 1022976t^{30} + 3561408t^{28} + 6023808t^{26} + 1874880t^{24} - 7796736t^{22} - 9577440t^{20} - 4719168t^{18} - 9577440t^{16} - 7796736t^{14} + 1874880t^{12} + 6023808t^{10} + 3561408t^{8} + 1022976t^{6} + 159408t^{4} + 12960t^{2} + 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 524685376x + 4626055605376$, with conductor $69360$
Generic density of odd order reductions $109/896$

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