Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $1600$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}57&112\\64&17\end{bmatrix}$, $\begin{bmatrix}61&10\\54&19\end{bmatrix}$, $\begin{bmatrix}149&116\\142&105\end{bmatrix}$, $\begin{bmatrix}181&28\\88&105\end{bmatrix}$, $\begin{bmatrix}215&164\\224&211\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.bk.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $192$ |
| Cyclic 240-torsion field degree: | $12288$ |
| Full 240-torsion field degree: | $5898240$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $5 y^{2} + 5 z^{2} + 4 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 5 y^{2} z^{2} + 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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{(25z^{4}+20z^{2}w^{2}+16w^{4})^{3}}{w^{4}z^{4}(5z^{2}+4w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.bk.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{5}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+5Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.48.0-8.f.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 240.48.0-8.f.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 240.192.3-80.be.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.be.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.ds.1.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.ds.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.fi.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.fi.2.4 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jo.1.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jo.2.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-120.bbc.1.14 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-120.ny.1.7 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-40.ci.1.1 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |