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: | 48.192.3.210 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&28\\26&3\end{bmatrix}$, $\begin{bmatrix}17&24\\40&1\end{bmatrix}$, $\begin{bmatrix}21&40\\10&17\end{bmatrix}$, $\begin{bmatrix}29&4\\40&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.bx.1 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 $ | $=$ | $ 4 x^{4} + 8 x^{3} z + 6 x^{2} z^{2} + 2 x z^{3} - y^{4} - 2 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^8}\cdot\frac{(256y^{8}+576y^{4}z^{4}+81z^{8})^{3}}{z^{16}y^{4}(4y^{4}+9z^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.p.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-48.b.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-48.b.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-48.b.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-48.b.2.11 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 48.96.1-8.p.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
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.ce.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ce.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.cg.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.cg.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ci.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ci.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ck.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-48.ck.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.576.19-48.kk.1.28 | $48$ | $3$ | $3$ | $19$ | $3$ | $1^{16}$ |
| 48.768.21-48.fz.1.20 | $48$ | $4$ | $4$ | $21$ | $3$ | $1^{18}$ |
| 240.384.5-240.ls.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ls.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lt.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lt.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lu.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lu.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lv.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.lv.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |