Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 16H0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}34&125\\121&238\end{bmatrix}$, $\begin{bmatrix}92&195\\129&142\end{bmatrix}$, $\begin{bmatrix}121&112\\164&181\end{bmatrix}$, $\begin{bmatrix}161&210\\168&211\end{bmatrix}$, $\begin{bmatrix}190&53\\127&228\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bs.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $24$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $5898240$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3}\cdot\frac{(x-y)^{48}(6561x^{16}+524880x^{14}y^{2}+1574640x^{12}y^{4}+1632960x^{10}y^{6}+1417824x^{8}y^{8}+725760x^{6}y^{10}+311040x^{4}y^{12}+46080x^{2}y^{14}+256y^{16})^{3}}{y^{2}x^{2}(x-y)^{48}(3x^{2}-2y^{2})^{16}(3x^{2}+2y^{2})^{4}(9x^{4}+4y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 80.48.0-16.g.1.2 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.e.2.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.e.2.11 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-16.g.1.7 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-24.by.1.15 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.192.1-48.j.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.s.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.bk.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.br.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.dn.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.dq.2.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.ec.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.eh.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbj.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbn.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbz.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcd.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcr.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcz.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdx.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bef.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-48.iu.1.22 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-48.hp.2.10 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.gm.1.20 | $240$ | $5$ | $5$ | $16$ |