Invariants
Level: | $240$ | $\SL_2$-level: | $16$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2^{3}\cdot16$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16D0 |
Level structure
$\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}78&17\\47&216\end{bmatrix}$, $\begin{bmatrix}86&209\\37&66\end{bmatrix}$, $\begin{bmatrix}98&127\\153&200\end{bmatrix}$, $\begin{bmatrix}148&25\\191&214\end{bmatrix}$, $\begin{bmatrix}153&178\\80&91\end{bmatrix}$, $\begin{bmatrix}229&206\\226&153\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.24.0.n.1 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $48$ |
Cyclic 240-torsion field degree: | $3072$ |
Full 240-torsion field degree: | $11796480$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.24.0-8.n.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
80.24.0-8.n.1.5 | $80$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.