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: | $\begin{bmatrix}23&176\\144&119\end{bmatrix}$, $\begin{bmatrix}65&176\\138&239\end{bmatrix}$, $\begin{bmatrix}83&184\\4&99\end{bmatrix}$, $\begin{bmatrix}135&208\\188&185\end{bmatrix}$, $\begin{bmatrix}171&80\\14&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.1.o.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| 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 $ | $=$ | $ x z + 2 y^{2} + z^{2} $ |
| $=$ | $6 x^{2} + 8 x z - 8 y^{2} + 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 y^{2} z^{2} + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{(1296z^{8}+432z^{6}w^{2}+180z^{4}w^{4}+24z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(6z^{2}+w^{2})^{4}(12z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.o.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+6Y^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 80.96.0-16.d.2.12 | $80$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 120.96.0-24.bc.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-16.d.2.3 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-24.bc.1.6 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.bl.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.bl.1.15 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.br.2.6 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.0-48.br.2.11 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.96.1-48.a.2.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.a.2.16 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bp.2.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bp.2.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bv.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-48.bv.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | 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.bh.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.by.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ee.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-48.ej.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bfs.2.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bfw.1.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bgq.1.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bgu.2.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |