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: | $\begin{bmatrix}21&28\\8&35\end{bmatrix}$, $\begin{bmatrix}85&196\\232&147\end{bmatrix}$, $\begin{bmatrix}89&28\\145&141\end{bmatrix}$, $\begin{bmatrix}131&60\\39&79\end{bmatrix}$, $\begin{bmatrix}209&64\\160&37\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 240.48.0.cf.1 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.1.15 | $80$ | $2$ | $2$ | $0$ | $?$ |
120.48.0-120.dj.1.23 | $120$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-48.e.1.24 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-80.n.1.14 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-120.dj.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.kl.1.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.km.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.lb.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.lc.1.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ub.1.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.uc.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.uz.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.va.1.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.baf.2.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bag.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bbd.2.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bbe.2.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bef.1.13 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.beg.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bev.1.14 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.bew.1.13 | $240$ | $2$ | $2$ | $1$ |
240.288.8-240.hd.1.35 | $240$ | $3$ | $3$ | $8$ |
240.384.7-240.tu.2.57 | $240$ | $4$ | $4$ | $7$ |
240.480.16-240.dh.2.28 | $240$ | $5$ | $5$ | $16$ |