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: | $10$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}22&189\\45&62\end{bmatrix}$, $\begin{bmatrix}46&231\\91&10\end{bmatrix}$, $\begin{bmatrix}89&130\\150&13\end{bmatrix}$, $\begin{bmatrix}104&131\\77&6\end{bmatrix}$, $\begin{bmatrix}105&104\\68&181\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.bi.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^2}\cdot\frac{(x+y)^{48}(1153x^{16}-2272x^{15}y+7344x^{14}y^{2}-38336x^{13}y^{3}+67952x^{12}y^{4}-169344x^{11}y^{5}+338752x^{10}y^{6}-318208x^{9}y^{7}+961632x^{8}y^{8}+636416x^{7}y^{9}+1355008x^{6}y^{10}+1354752x^{5}y^{11}+1087232x^{4}y^{12}+1226752x^{3}y^{13}+470016x^{2}y^{14}+290816xy^{15}+295168y^{16})^{3}}{(x+y)^{48}(x^{2}+2y^{2})^{4}(x^{2}-2xy-2y^{2})^{2}(x^{2}+4xy-2y^{2})^{16}(x^{4}+2x^{3}y+6x^{2}y^{2}-4xy^{3}+4y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 80.48.0-16.f.2.10 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-16.f.2.8 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.g.1.8 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.g.1.19 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-24.by.1.9 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.p.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.x.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.bq.2.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.bv.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.cm.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.cp.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.db.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-48.dg.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wr.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wz.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xx.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yf.1.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zd.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zl.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.baj.1.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bar.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-48.ia.1.21 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-48.gv.2.24 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.fu.1.16 | $240$ | $5$ | $5$ | $16$ |