Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot4\cdot16$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16C0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}9&218\\224&159\end{bmatrix}$, $\begin{bmatrix}21&182\\52&155\end{bmatrix}$, $\begin{bmatrix}44&113\\193&168\end{bmatrix}$, $\begin{bmatrix}173&222\\16&55\end{bmatrix}$, $\begin{bmatrix}194&31\\117&236\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.24.0.i.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $96$ |
| Cyclic 240-torsion field degree: | $6144$ |
| Full 240-torsion field degree: | $11796480$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 58 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^8}\cdot\frac{x^{24}(81x^{8}+576x^{4}y^{4}+256y^{8})^{3}}{y^{4}x^{40}(9x^{4}+4y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-8.o.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 240.24.0-8.o.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.96.1-48.b.1.17 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.c.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.n.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.o.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.z.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.ba.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.bd.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.be.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gw.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gx.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.ha.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.hb.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.hm.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.hn.1.16 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.hq.1.16 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.hr.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.144.4-48.bi.1.21 | $240$ | $3$ | $3$ | $4$ |
| 240.192.3-48.qh.1.13 | $240$ | $4$ | $4$ | $3$ |
| 240.240.8-240.ba.1.28 | $240$ | $5$ | $5$ | $8$ |
| 240.288.7-240.yo.1.29 | $240$ | $6$ | $6$ | $7$ |
| 240.480.15-240.cc.1.25 | $240$ | $10$ | $10$ | $15$ |