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 | $4^{2}$ | ||
| 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: | $\begin{bmatrix}51&52\\65&57\end{bmatrix}$, $\begin{bmatrix}113&114\\15&151\end{bmatrix}$, $\begin{bmatrix}143&10\\204&137\end{bmatrix}$, $\begin{bmatrix}173&84\\230&49\end{bmatrix}$, $\begin{bmatrix}223&70\\25&201\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.1.ic.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} - x y - y^{2} - z^{2} $ |
| $=$ | $2 x^{2} + 3 x y + 3 y^{2} - 5 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 70 x^{2} y^{2} - 12 x^{2} z^{2} + 25 y^{4} + 30 y^{2} z^{2} + 4 z^{4} $ |
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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(2z^{2}+w^{2})^{3}(6z^{2}-w^{2})^{3}}{z^{4}(2z^{2}-w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.48.1.ic.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-70X^{2}Y^{2}+25Y^{4}-12X^{2}Z^{2}+30Y^{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 |
|---|---|---|---|---|---|---|
| 240.48.0-40.ci.1.4 | $240$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 240.48.0-40.ci.1.8 | $240$ | $2$ | $2$ | $0$ | $?$ | 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.wb.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.wc.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.wd.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.we.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.exj.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.exk.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.exl.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.exm.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.288.9-120.bexm.1.15 | $240$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 240.384.9-120.dnv.1.13 | $240$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 240.480.17-40.xy.1.5 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |