Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $1152$ | ||
| 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 $4$ are rational) | Cusp widths | $8^{8}\cdot16^{8}\cdot24^{8}\cdot48^{8}$ | Cusp orbits | $1^{4}\cdot2^{6}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.1536.49.87068 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&42\\36&25\end{bmatrix}$, $\begin{bmatrix}25&10\\0&35\end{bmatrix}$, $\begin{bmatrix}25&12\\36&47\end{bmatrix}$, $\begin{bmatrix}41&20\\24&37\end{bmatrix}$, $\begin{bmatrix}47&16\\12&17\end{bmatrix}$ |
| $\GL_2(\Z/48\Z)$-subgroup: | $D_6\times D_8:C_4$ |
| Contains $-I$: | no $\quad$ (see 48.768.49.rg.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{279}\cdot3^{63}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{6}\cdot12^{2}$ |
| Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 32.2.a.a$^{2}$, 48.2.k.a, 96.2.a.a, 96.2.a.b, 384.2.k.b, 1152.2.a.a, 1152.2.a.b, 1152.2.a.i, 1152.2.a.l, 1152.2.a.m$^{2}$, 1152.2.a.r$^{2}$, 1152.2.d.a, 1152.2.d.d$^{2}$, 1152.2.d.f |
Rational points
This modular curve has 4 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.ef.1.16 | $48$ | $2$ | $2$ | $21$ | $0$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.768.21-48.ef.1.46 | $48$ | $2$ | $2$ | $21$ | $0$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.768.21-48.gb.1.30 | $48$ | $2$ | $2$ | $21$ | $0$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.768.21-48.gb.1.48 | $48$ | $2$ | $2$ | $21$ | $0$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.768.25-48.hy.1.26 | $48$ | $2$ | $2$ | $25$ | $4$ | $12^{2}$ |
| 48.768.25-48.hy.1.29 | $48$ | $2$ | $2$ | $25$ | $4$ | $12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.3072.97-48.iw.2.1 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.iy.4.5 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.jc.1.3 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.je.1.7 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.oj.3.16 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.or.1.14 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.sx.2.24 | $48$ | $2$ | $2$ | $97$ | $6$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.td.2.16 | $48$ | $2$ | $2$ | $97$ | $6$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.tt.2.16 | $48$ | $2$ | $2$ | $97$ | $8$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.tu.4.16 | $48$ | $2$ | $2$ | $97$ | $8$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.bbp.2.12 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.bbx.2.16 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.bmz.2.15 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.bnh.2.11 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.brm.2.22 | $48$ | $2$ | $2$ | $97$ | $6$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.brt.1.14 | $48$ | $2$ | $2$ | $97$ | $6$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.bsj.1.14 | $48$ | $2$ | $2$ | $97$ | $8$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.bsk.1.12 | $48$ | $2$ | $2$ | $97$ | $8$ | $1^{12}\cdot2^{4}\cdot4\cdot12^{2}$ |
| 48.3072.97-48.caf.1.13 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.can.3.15 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{6}\cdot4^{5}\cdot8^{2}$ |
| 48.3072.97-48.cdy.1.3 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.cea.1.7 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.cee.2.1 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.3072.97-48.ceg.4.5 | $48$ | $2$ | $2$ | $97$ | $4$ | $2^{8}\cdot4^{4}\cdot8^{2}$ |
| 48.4608.161-48.clh.1.19 | $48$ | $3$ | $3$ | $161$ | $12$ | $1^{28}\cdot2^{14}\cdot12^{2}\cdot16^{2}$ |