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: | $\begin{bmatrix}3&68\\107&79\end{bmatrix}$, $\begin{bmatrix}3&236\\50&37\end{bmatrix}$, $\begin{bmatrix}91&128\\78&191\end{bmatrix}$, $\begin{bmatrix}161&92\\66&239\end{bmatrix}$, $\begin{bmatrix}219&52\\214&121\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.48.0.cd.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 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 |
---|---|---|---|---|---|
48.48.0-16.f.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ |
80.48.0-16.f.2.1 | $80$ | $2$ | $2$ | $0$ | $?$ |
120.48.0-120.dh.1.15 | $120$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.n.1.13 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.n.1.22 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-120.dh.1.11 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.kh.2.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ki.2.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.kx.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ky.2.5 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.un.2.5 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.uo.2.9 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.uv.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.uw.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bar.2.11 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bas.2.4 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.baz.2.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bba.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.beb.2.5 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bec.2.3 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ber.2.11 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bes.2.7 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.hb.1.12 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.ts.1.10 | $240$ | $4$ | $4$ | $7$ |
240.480.16-240.df.2.9 | $240$ | $5$ | $5$ | $16$ |