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 | $2^{5}$ | ||
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: | $\begin{bmatrix}41&208\\106&169\end{bmatrix}$, $\begin{bmatrix}73&148\\49&41\end{bmatrix}$, $\begin{bmatrix}213&44\\107&89\end{bmatrix}$, $\begin{bmatrix}215&144\\149&73\end{bmatrix}$, $\begin{bmatrix}235&116\\138&221\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.48.0.ce.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 |
---|---|---|---|---|---|
16.48.0-16.e.1.15 | $16$ | $2$ | $2$ | $0$ | $0$ |
120.48.0-120.dj.1.8 | $120$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-16.e.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.n.1.39 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.n.1.56 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-120.dj.1.4 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.kj.2.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.kk.2.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.kz.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.la.2.5 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.tz.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ua.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ux.2.6 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.uy.2.7 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bad.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bae.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bbb.2.6 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bbc.2.7 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bed.2.16 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bee.2.16 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bet.2.8 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.beu.2.8 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.hc.1.51 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.tt.2.15 | $240$ | $4$ | $4$ | $7$ |
240.480.16-240.dg.2.8 | $240$ | $5$ | $5$ | $16$ |