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: | $2$ | ||||||
$\overline{\Q}$-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: | $\begin{bmatrix}37&56\\211&231\end{bmatrix}$, $\begin{bmatrix}95&8\\179&69\end{bmatrix}$, $\begin{bmatrix}187&200\\37&213\end{bmatrix}$, $\begin{bmatrix}191&96\\102&11\end{bmatrix}$, $\begin{bmatrix}201&8\\128&153\end{bmatrix}$, $\begin{bmatrix}215&136\\81&169\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.0.j.1 for the level structure with $-I$) |
Cyclic 240-isogeny field degree: | $48$ |
Cyclic 240-torsion field degree: | $3072$ |
Full 240-torsion field degree: | $5898240$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 12 x^{2} + 2 y^{2} + 3 z^{2} $ |
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 |
---|---|---|---|---|---|
40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
240.48.0-48.e.1.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-48.e.1.30 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-48.e.2.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-48.e.2.31 | $240$ | $2$ | $2$ | $0$ | $?$ |
240.48.0-8.q.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.