Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $384$ | ||
| Index: | $1536$ | $\PSL_2$-index: | $768$ | ||||
| Genus: | $49 = 1 + \frac{ 768 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $8$ are rational) | Cusp widths | $8^{8}\cdot16^{8}\cdot24^{8}\cdot48^{8}$ | Cusp orbits | $1^{8}\cdot2^{4}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $8 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $8 \le \gamma \le 24$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.1536.49.72382 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}17&14\\24&23\end{bmatrix}$, $\begin{bmatrix}17&16\\0&37\end{bmatrix}$, $\begin{bmatrix}23&30\\36&7\end{bmatrix}$, $\begin{bmatrix}31&40\\36&25\end{bmatrix}$, $\begin{bmatrix}47&8\\12&1\end{bmatrix}$ |
| $\GL_2(\Z/48\Z)$-subgroup: | $D_6\times D_8:C_4$ |
| Contains $-I$: | no $\quad$ (see 48.768.49.biu.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{279}\cdot3^{43}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| Newforms: | 16.2.e.a$^{2}$, 24.2.a.a, 24.2.d.a$^{2}$, 32.2.a.a$^{2}$, 48.2.j.a, 96.2.a.a, 96.2.a.b, 384.2.c.b, 384.2.c.c, 384.2.f.b, 384.2.f.d, 384.2.k.b |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.21-48.gb.1.25 | $48$ | $2$ | $2$ | $21$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.768.21-48.gb.1.48 | $48$ | $2$ | $2$ | $21$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.768.21-48.gg.1.40 | $48$ | $2$ | $2$ | $21$ | $0$ | $4^{4}\cdot12$ |
| 48.768.21-48.gg.1.41 | $48$ | $2$ | $2$ | $21$ | $0$ | $4^{4}\cdot12$ |
| 48.768.25-48.hu.1.25 | $48$ | $2$ | $2$ | $25$ | $0$ | $2^{2}\cdot8\cdot12$ |
| 48.768.25-48.hu.1.33 | $48$ | $2$ | $2$ | $25$ | $0$ | $2^{2}\cdot8\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.3072.97-48.ty.2.15 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.zf.1.13 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.bbw.1.15 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.bck.2.13 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.brr.2.2 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.brx.4.2 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.byq.1.2 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.byr.1.1 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.caj.1.1 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cak.1.2 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.cbe.2.2 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.cbf.1.5 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cee.2.1 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.cef.1.7 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cep.3.5 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cer.3.1 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.cfi.1.1 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.cfj.1.5 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cgt.1.11 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cgw.4.1 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{10}\cdot8\cdot12$ |
| 48.3072.97-48.chz.2.9 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.cif.1.13 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.cip.1.9 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.3072.97-48.cir.2.13 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{6}\cdot4^{2}\cdot8\cdot12$ |
| 48.4608.161-48.dcx.1.29 | $48$ | $3$ | $3$ | $161$ | $2$ | $1^{12}\cdot2^{8}\cdot4^{8}\cdot8^{3}\cdot12\cdot16$ |