Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{8}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.1242 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&10\\40&37\end{bmatrix}$, $\begin{bmatrix}23&24\\12&1\end{bmatrix}$, $\begin{bmatrix}29&46\\24&25\end{bmatrix}$, $\begin{bmatrix}31&28\\28&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.cb.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{16}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 64.2.a.a, 288.2.a.d$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{4} + 2 y^{4} - 2 y^{3} z - 6 y^{2} z^{2} - 8 y z^{3} - 4 z^{4} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,29$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{(y^{4}+2y^{3}z+6y^{2}z^{2}-4yz^{3}+4z^{4})^{3}(13y^{4}+38y^{3}z+42y^{2}z^{2}+20yz^{3}+4z^{4})^{3}}{y^{4}(y+2z)^{4}(y^{2}-2yz-2z^{2})^{4}(y^{2}+yz+z^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.a.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.bu.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-16.a.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-48.b.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-48.b.1.19 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-24.bu.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-48.bb.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.bb.2.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.bj.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.bj.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.dz.1.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.dz.2.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ef.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ef.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.576.19-48.lx.1.12 | $48$ | $3$ | $3$ | $19$ | $3$ | $1^{16}$ |
| 48.768.21-48.je.1.17 | $48$ | $4$ | $4$ | $21$ | $3$ | $1^{18}$ |
| 240.384.5-240.sf.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.sf.2.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.sj.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.sj.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wl.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wl.2.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wt.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wt.2.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |