Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $2304$ | ||
| Index: | $768$ | $\PSL_2$-index: | $768$ | ||||
| Genus: | $57 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $48^{16}$ | Cusp orbits | $16$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $21$ | ||||||
| $\Q$-gonality: | $12 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $12 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.57.7 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}19&30\\42&37\end{bmatrix}$, $\begin{bmatrix}21&29\\47&40\end{bmatrix}$, $\begin{bmatrix}23&0\\0&23\end{bmatrix}$, $\begin{bmatrix}37&12\\36&25\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 48.1536.57-48.a.1.1, 48.1536.57-48.a.1.2, 48.1536.57-48.a.1.3, 48.1536.57-48.a.1.4 |
| Cyclic 48-isogeny field degree: | $96$ |
| Cyclic 48-torsion field degree: | $1536$ |
| Full 48-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{422}\cdot3^{100}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{33}\cdot2^{10}\cdot4$ |
| Newforms: | 36.2.a.a, 64.2.a.a, 144.2.a.a, 144.2.a.b, 256.2.a.a, 256.2.a.b, 256.2.a.c, 256.2.a.d, 256.2.a.e, 576.2.a.a, 576.2.a.b, 576.2.a.c, 576.2.a.d, 576.2.a.e, 576.2.a.f, 576.2.a.g, 576.2.a.h, 576.2.a.i, 2304.2.a.a, 2304.2.a.b, 2304.2.a.c, 2304.2.a.d, 2304.2.a.e, 2304.2.a.f, 2304.2.a.g, 2304.2.a.h, 2304.2.a.i, 2304.2.a.j, 2304.2.a.k, 2304.2.a.l, 2304.2.a.m, 2304.2.a.n, 2304.2.a.o, 2304.2.a.p, 2304.2.a.q, 2304.2.a.r, 2304.2.a.s, 2304.2.a.t, 2304.2.a.u, 2304.2.a.v, 2304.2.a.w, 2304.2.a.x, 2304.2.a.y, 2304.2.a.z |
Rational points
This modular curve has no real points, and therefore no 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(3)$ | $3$ | $128$ | $128$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}(16)$ | $16$ | $6$ | $6$ | $7$ | $2$ | $1^{28}\cdot2^{9}\cdot4$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(24)$ | $24$ | $4$ | $4$ | $13$ | $3$ | $1^{20}\cdot2^{10}\cdot4$ |
| $X_{\mathrm{ns}}^+(48)$ | $48$ | $2$ | $2$ | $21$ | $21$ | $1^{20}\cdot2^{6}\cdot4$ |
| 48.384.25.cdu.1 | $48$ | $2$ | $2$ | $25$ | $9$ | $1^{18}\cdot2^{5}\cdot4$ |
| 48.384.29.a.1 | $48$ | $2$ | $2$ | $29$ | $9$ | $1^{18}\cdot2^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.2304.169.a.2 | $48$ | $3$ | $3$ | $169$ | $63$ | $1^{64}\cdot2^{20}\cdot4^{2}$ |