Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $3^{4}\cdot12\cdot48$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48B4 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}17&174\\12&95\end{bmatrix}$, $\begin{bmatrix}73&52\\218&175\end{bmatrix}$, $\begin{bmatrix}112&13\\175&14\end{bmatrix}$, $\begin{bmatrix}113&234\\12&107\end{bmatrix}$, $\begin{bmatrix}130&57\\93&154\end{bmatrix}$, $\begin{bmatrix}219&236\\160&183\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.72.4.bi.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $96$ |
| Cyclic 240-torsion field degree: | $6144$ |
| Full 240-torsion field degree: | $3932160$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 6 x^{2} - y w $ |
| $=$ | $x y^{2} + 16 x w^{2} - 6 z^{3}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 36 x^{5} - x z^{4} + 12 y^{3} z^{2} $ |
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 |
|---|
| $(0:1:0:0)$, $(0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{4}+16y^{2}w^{2}+16w^{4})^{3}}{w^{8}y^{2}(y^{2}+16w^{2})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.72.4.bi.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ -36X^{5}+12Y^{3}Z^{2}-XZ^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 120.72.2-24.cw.1.14 | $120$ | $2$ | $2$ | $2$ | $?$ |
| 240.48.0-48.i.1.12 | $240$ | $3$ | $3$ | $0$ | $?$ |
| 240.72.2-24.cw.1.15 | $240$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.