Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $384$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $21 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot8^{6}\cdot12^{4}\cdot16^{2}\cdot24^{6}\cdot48^{2}$ | Cusp orbits | $1^{4}\cdot2^{6}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 12$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48CP21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.768.21.35288 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&2\\24&11\end{bmatrix}$, $\begin{bmatrix}23&2\\36&43\end{bmatrix}$, $\begin{bmatrix}31&8\\36&37\end{bmatrix}$, $\begin{bmatrix}47&10\\12&7\end{bmatrix}$, $\begin{bmatrix}47&18\\12&47\end{bmatrix}$, $\begin{bmatrix}47&28\\24&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.384.21.gb.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $32$ |
| Full 48-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{119}\cdot3^{19}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}\cdot12$ |
| Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 32.2.a.a$^{2}$, 96.2.a.a, 96.2.a.b, 384.2.k.b |
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 |
|---|---|---|---|---|---|---|
| 48.384.9-24.by.1.16 | $48$ | $2$ | $2$ | $9$ | $0$ | $12$ |
| 48.384.9-24.by.1.28 | $48$ | $2$ | $2$ | $9$ | $0$ | $12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.1536.41-48.fx.1.21 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.ga.2.23 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.gj.2.22 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.gm.1.21 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.jc.1.16 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.jf.1.13 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.jo.1.15 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.jr.1.15 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.le.1.24 | $48$ | $2$ | $2$ | $41$ | $2$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.lf.1.22 | $48$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.na.3.36 | $48$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.nl.2.23 | $48$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.rj.2.27 | $48$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.rk.1.21 | $48$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.rw.1.22 | $48$ | $2$ | $2$ | $41$ | $2$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.rx.1.18 | $48$ | $2$ | $2$ | $41$ | $0$ | $1^{4}\cdot2^{2}\cdot12$ |
| 48.1536.41-48.tf.1.23 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.th.1.23 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.tn.1.21 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.tp.1.24 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{6}\cdot8$ |
| 48.1536.41-48.um.1.19 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.un.2.20 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.uq.4.23 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.41-48.ur.1.18 | $48$ | $2$ | $2$ | $41$ | $0$ | $2^{4}\cdot4\cdot8$ |
| 48.1536.49-48.ms.1.18 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.mt.1.8 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.mu.1.5 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.mv.2.24 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.na.2.20 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.nb.1.7 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.nc.1.7 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.nd.1.20 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.re.1.9 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.rf.1.11 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.rg.2.31 | $48$ | $2$ | $2$ | $49$ | $4$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.rh.1.31 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bac.4.26 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bao.4.18 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bbb.2.21 | $48$ | $2$ | $2$ | $49$ | $4$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.bbf.1.23 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.bei.4.20 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bej.1.28 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bek.1.5 | $48$ | $2$ | $2$ | $49$ | $4$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.bel.1.7 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.bho.1.1 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.bht.1.3 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{2}\cdot4\cdot12$ |
| 48.1536.49-48.bia.2.32 | $48$ | $2$ | $2$ | $49$ | $4$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bib.1.32 | $48$ | $2$ | $2$ | $49$ | $2$ | $1^{8}\cdot2^{4}\cdot12$ |
| 48.1536.49-48.bik.1.23 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.bil.1.19 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.bim.2.18 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.bin.1.23 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.bis.1.21 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.bit.2.22 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.biu.1.17 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.1536.49-48.biv.1.24 | $48$ | $2$ | $2$ | $49$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.2304.73-48.oc.1.43 | $48$ | $3$ | $3$ | $73$ | $2$ | $1^{12}\cdot2^{6}\cdot12\cdot16$ |