Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}41&152\\43&135\end{bmatrix}$, $\begin{bmatrix}199&96\\192&55\end{bmatrix}$, $\begin{bmatrix}209&184\\86&239\end{bmatrix}$, $\begin{bmatrix}231&200\\40&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.1.ds.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $24$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $2949120$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 x^{2} + 5 y^{2} - z^{2} - w^{2} $ |
| $=$ | $8 x^{2} - 2 y^{2} + z^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3}\cdot\frac{(81z^{8}+3240z^{6}w^{2}+4824z^{4}w^{4}+1440z^{2}w^{6}+16w^{8})^{3}}{w^{2}z^{2}(3z^{2}-2w^{2})^{8}(3z^{2}+2w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.96.0-16.x.2.2 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.bl.2.2 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-16.x.2.2 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.x.2.8 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.x.2.12 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-24.bl.2.6 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.bu.1.5 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.bu.1.14 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.1-48.bt.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bt.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bu.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bu.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.cf.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.cf.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |