Invariants
Level: | $48$ | $\SL_2$-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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1153 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&11\\0&35\end{bmatrix}$, $\begin{bmatrix}25&14\\44&39\end{bmatrix}$, $\begin{bmatrix}29&5\\12&1\end{bmatrix}$, $\begin{bmatrix}31&45\\24&29\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 24.48.0.bk.2 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $64$ |
Full 48-torsion field degree: | $12288$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} + 12 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 |
---|---|---|---|---|---|
16.48.0-8.ba.2.6 | $16$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-8.ba.2.6 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-24.bj.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-24.bj.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-24.bz.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-24.bz.1.11 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
48.192.1-48.cj.2.2 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cl.2.4 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cr.1.6 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.ct.1.2 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dp.1.5 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dr.1.6 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dx.2.6 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dz.2.2 | $48$ | $2$ | $2$ | $1$ |
48.288.8-24.gk.1.15 | $48$ | $3$ | $3$ | $8$ |
48.384.7-24.el.1.6 | $48$ | $4$ | $4$ | $7$ |
240.192.1-240.mb.1.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.md.2.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.mj.2.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ml.1.4 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qz.1.10 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rb.1.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rh.2.12 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rj.2.4 | $240$ | $2$ | $2$ | $1$ |
240.480.16-120.es.1.6 | $240$ | $5$ | $5$ | $16$ |