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 | $2^{8}\cdot16^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16G0 |
Level structure
$\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}75&236\\154&213\end{bmatrix}$, $\begin{bmatrix}97&20\\90&71\end{bmatrix}$, $\begin{bmatrix}173&4\\93&185\end{bmatrix}$, $\begin{bmatrix}195&128\\23&145\end{bmatrix}$, $\begin{bmatrix}205&196\\133&29\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.48.0.bt.2 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $48$ |
Cyclic 240-torsion field degree: | $3072$ |
Full 240-torsion field degree: | $5898240$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
48.48.0-48.e.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-80.n.2.25 | $80$ | $2$ | $2$ | $0$ | $?$ |
120.48.0-120.dd.1.24 | $120$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-48.e.1.21 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-80.n.2.13 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-120.dd.1.12 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
240.192.1-240.ix.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.iy.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.jn.1.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.jo.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.td.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.te.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.tl.1.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.tm.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.zh.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.zi.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.zp.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.zq.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bcr.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bcs.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bdh.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bdi.1.14 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.gf.2.37 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.tg.1.47 | $240$ | $4$ | $4$ | $7$ |
240.480.16-240.cv.1.28 | $240$ | $5$ | $5$ | $16$ |