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 | $4^{8}\cdot8^{8}$ | 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: | 8K1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}33&128\\164&77\end{bmatrix}$, $\begin{bmatrix}115&116\\86&237\end{bmatrix}$, $\begin{bmatrix}123&68\\20&233\end{bmatrix}$, $\begin{bmatrix}139&28\\100&53\end{bmatrix}$, $\begin{bmatrix}205&228\\8&139\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.cf.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| 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 $ | $=$ | $ 2 x^{2} - 2 y^{2} - z^{2} $ |
| $=$ | $3 x^{2} + 3 y^{2} - w^{2}$ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^2}\cdot\frac{(81z^{8}+504z^{4}w^{4}+16w^{8})^{3}}{w^{4}z^{4}(3z^{2}-2w^{2})^{4}(3z^{2}+2w^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.96.0-8.k.1.1 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-8.k.1.3 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-24.ba.2.1 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-24.ba.2.5 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.1-24.bv.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-24.bv.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 240.384.5-48.bc.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.bc.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ch.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ch.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ec.2.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ec.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.em.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.em.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.tn.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.tn.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.tz.1.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.tz.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xv.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xv.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.yh.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.yh.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |