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 | $2^{8}\cdot16^{2}$ | 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 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.730 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&35\\8&3\end{bmatrix}$, $\begin{bmatrix}1&47\\0&23\end{bmatrix}$, $\begin{bmatrix}5&37\\20&41\end{bmatrix}$, $\begin{bmatrix}15&11\\32&17\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 48.48.0.u.2 for the level structure with $-I$) |
Cyclic 48-isogeny field degree: | $8$ |
Cyclic 48-torsion field degree: | $32$ |
Full 48-torsion field degree: | $12288$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 3 x^{2} + 12 x y - 12 y^{2} - z^{2} $ |
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.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.bh.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-16.e.1.16 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.f.2.8 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-48.f.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
48.48.0-24.bh.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
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.cf.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cg.1.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cn.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.co.2.2 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dl.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dm.2.2 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dt.2.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.du.1.1 | $48$ | $2$ | $2$ | $1$ |
48.288.8-48.dc.1.1 | $48$ | $3$ | $3$ | $8$ |
48.384.7-48.eb.1.1 | $48$ | $4$ | $4$ | $7$ |
240.192.1-240.lj.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.lk.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.lr.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ls.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qh.1.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qi.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qp.2.1 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qq.1.1 | $240$ | $2$ | $2$ | $1$ |
240.480.16-240.bu.1.2 | $240$ | $5$ | $5$ | $16$ |