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: | $32$ (none of which are rational) | Cusp widths | $8^{8}\cdot16^{8}\cdot24^{8}\cdot48^{8}$ | Cusp orbits | $2^{8}\cdot4^{4}$ | ||
| 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: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.1536.49.80152 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&6\\24&11\end{bmatrix}$, $\begin{bmatrix}7&4\\12&25\end{bmatrix}$, $\begin{bmatrix}7&12\\36&13\end{bmatrix}$, $\begin{bmatrix}7&40\\24&11\end{bmatrix}$, $\begin{bmatrix}41&26\\24&19\end{bmatrix}$ |
| $\GL_2(\Z/48\Z)$-subgroup: | $D_6\times D_8:C_4$ |
| Contains $-I$: | no $\quad$ (see 48.768.49.bin.1 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: | $2^{279}\cdot3^{59}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 32.2.a.a$^{2}$, 96.2.a.a, 96.2.a.b, 144.2.k.a$^{2}$, 144.2.k.b, 384.2.c.a, 384.2.c.d, 384.2.f.a, 384.2.f.c, 384.2.k.b |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.1.38 | $48$ | $2$ | $2$ | $21$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.768.21-48.gb.1.48 | $48$ | $2$ | $2$ | $21$ | $0$ | $2^{2}\cdot4^{4}\cdot8$ |
| 48.768.21-48.ge.1.23 | $48$ | $2$ | $2$ | $21$ | $0$ | $4^{4}\cdot12$ |
| 48.768.21-48.ge.1.55 | $48$ | $2$ | $2$ | $21$ | $0$ | $4^{4}\cdot12$ |
| 48.768.25-48.hx.2.31 | $48$ | $2$ | $2$ | $25$ | $0$ | $2^{2}\cdot8\cdot12$ |
| 48.768.25-48.hx.2.35 | $48$ | $2$ | $2$ | $25$ | $0$ | $2^{2}\cdot8\cdot12$ |
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.or.1.14 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.oy.3.11 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.tx.2.14 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.yk.2.11 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.bna.2.5 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.bnc.2.7 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.bnw.1.5 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.bob.1.8 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.brq.2.5 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.bsg.1.7 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.byi.1.5 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.byn.1.6 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cbs.1.5 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.cbx.1.6 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.ccz.2.10 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cdh.1.11 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.cdw.1.5 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.ceb.1.8 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cep.3.5 | $48$ | $2$ | $2$ | $97$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{4}\cdot8\cdot12$ |
| 48.3072.97-48.cet.3.11 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.chm.2.7 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.chr.2.15 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.chy.4.11 | $48$ | $2$ | $2$ | $97$ | $2$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.3072.97-48.cid.1.14 | $48$ | $2$ | $2$ | $97$ | $4$ | $1^{8}\cdot2^{8}\cdot4\cdot8\cdot12$ |
| 48.4608.161-48.bna.1.5 | $48$ | $3$ | $3$ | $161$ | $2$ | $1^{12}\cdot2^{8}\cdot4^{8}\cdot8^{3}\cdot12\cdot16$ |