Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | Newform level: | $1152$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{6}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O3 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}7&18\\108&223\end{bmatrix}$, $\begin{bmatrix}33&214\\100&217\end{bmatrix}$, $\begin{bmatrix}59&40\\204&193\end{bmatrix}$, $\begin{bmatrix}93&28\\152&49\end{bmatrix}$, $\begin{bmatrix}193&176\\136&225\end{bmatrix}$, $\begin{bmatrix}237&118\\188&169\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.bf.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $96$ |
| Cyclic 240-torsion field degree: | $6144$ |
| Full 240-torsion field degree: | $2949120$ |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 18 x^{4} + 3 x^{2} y^{2} + 3 x^{2} z^{2} - y^{3} z - 2 y^{2} z^{2} + y z^{3} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{8}+8y^{7}z+20y^{6}z^{2}+8y^{5}z^{3}+230y^{4}z^{4}-8y^{3}z^{5}+20y^{2}z^{6}-8yz^{7}+z^{8})^{3}}{z^{4}y^{4}(y^{2}+2yz-z^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.96.1-8.k.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ |
| 240.96.1-8.k.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.