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: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-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: | $\begin{bmatrix}65&104\\212&85\end{bmatrix}$, $\begin{bmatrix}197&56\\11&27\end{bmatrix}$, $\begin{bmatrix}201&8\\163&191\end{bmatrix}$, $\begin{bmatrix}201&20\\50&43\end{bmatrix}$, $\begin{bmatrix}203&196\\12&77\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.48.0.bp.1 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $48$ |
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 real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.f.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-40.bp.1.7 | $80$ | $2$ | $2$ | $0$ | $?$ |
120.48.0-40.bp.1.9 | $120$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-48.f.1.21 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.m.1.34 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.m.1.51 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.il.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.im.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.it.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.iu.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.pn.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.po.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qd.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qe.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.yd.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ye.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.yt.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.yu.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bcf.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bcg.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bcn.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bco.1.1 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.fu.1.39 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.sr.2.9 | $240$ | $4$ | $4$ | $7$ |
240.480.16-240.cr.2.17 | $240$ | $5$ | $5$ | $16$ |