Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $13 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (of which $4$ are rational) | Cusp widths | $1^{8}\cdot2^{4}\cdot3^{8}\cdot4^{4}\cdot6^{4}\cdot12^{4}\cdot16^{4}\cdot48^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4^{7}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AH13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.13.5459 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&30\\24&1\end{bmatrix}$, $\begin{bmatrix}31&11\\0&1\end{bmatrix}$, $\begin{bmatrix}41&44\\0&1\end{bmatrix}$, $\begin{bmatrix}43&29\\12&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.13.px.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $8$ |
| Full 48-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{55}\cdot3^{13}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{5}$ |
| Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 24.2.f.a, 96.2.f.a, 192.2.a.b, 192.2.a.d, 192.2.c.a |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $96$ | $96$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.z.1.2 | $16$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.gk.3.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-24.gk.3.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-48.km.4.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-48.km.4.13 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.384.5-48.lb.4.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{4}$ |
| 48.384.5-48.lb.4.28 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{4}$ |
| 48.384.7-48.ff.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.ff.4.14 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.fu.4.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.fu.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.384.7-48.hw.1.8 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 48.384.7-48.hw.1.16 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 48.384.7-48.if.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
| 48.384.7-48.if.4.26 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.1536.33-48.ev.2.3 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.ik.2.8 | $48$ | $2$ | $2$ | $33$ | $0$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.lf.1.8 | $48$ | $2$ | $2$ | $33$ | $1$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.np.1.12 | $48$ | $2$ | $2$ | $33$ | $0$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.tz.2.3 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.uv.1.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.xb.1.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.1536.33-48.xl.1.4 | $48$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 48.2304.57-48.gr.1.4 | $48$ | $3$ | $3$ | $57$ | $2$ | $1^{10}\cdot2^{13}\cdot4^{2}$ |