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: | $10$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16H0 |
Level structure
$\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}61&16\\234&91\end{bmatrix}$, $\begin{bmatrix}63&118\\32&61\end{bmatrix}$, $\begin{bmatrix}76&119\\115&168\end{bmatrix}$, $\begin{bmatrix}88&171\\105&34\end{bmatrix}$, $\begin{bmatrix}209&92\\68&225\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.48.0.cj.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 infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
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.ej.1.28 | $120$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-16.e.1.12 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.o.1.62 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-240.o.1.63 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-120.ej.1.30 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.cb.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.cm.2.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.do.2.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.fo.2.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.jd.2.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.jm.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.kk.2.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.kr.2.3 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ls.2.9 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.lz.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.mx.2.11 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ng.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.od.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.om.2.5 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.pk.2.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.pr.2.11 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.sp.2.38 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.ya.1.62 | $240$ | $4$ | $4$ | $7$ |
240.480.16-240.dl.2.4 | $240$ | $5$ | $5$ | $16$ |