The modular curve $X_{10d}$

Curve name $X_{10d}$
Index $12$
Level $4$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
Meaning/Special name
Chosen covering $X_{10}$
Curves that $X_{10d}$ minimally covers
Curves that minimally cover $X_{10d}$
Curves that minimally cover $X_{10d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 81t^{4} - 27t^{2}\] \[B(t) = 486t^{5} + 54t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 13x + 156$, with conductor $130$
Generic density of odd order reductions $25/112$

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