The modular curve $X_{151}$

Curve name $X_{151}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 11 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
$8$ $12$ $X_{45}$
Meaning/Special name
Chosen covering $X_{45}$
Curves that $X_{151}$ minimally covers $X_{45}$
Curves that minimally cover $X_{151}$ $X_{310}$, $X_{321}$, $X_{363}$, $X_{373}$, $X_{375}$, $X_{378}$, $X_{382}$, $X_{387}$, $X_{393}$, $X_{396}$
Curves that minimally cover $X_{151}$ and have infinitely many rational points.
Model \[y^2 = x^3 - x^2 - 3x - 1\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[ \infty \]
$(-1 : 0 : 1)$ \[1728 \,\,(\text{CM by }-4)\]
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

Back to the 2-adic image homepage.