Invariants
Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
Index: | $384$ | $\PSL_2$-index: | $384$ | ||||
Genus: | $17 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
Cusps: | $32$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot6^{8}\cdot12^{4}\cdot16^{4}\cdot48^{4}$ | Cusp orbits | $2^{8}\cdot4^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $3$ | ||||||
$\Q$-gonality: | $6 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $6 \le \gamma \le 8$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 48CM17 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.17.1948 |
Level structure
$\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&11\\0&29\end{bmatrix}$, $\begin{bmatrix}17&38\\0&31\end{bmatrix}$, $\begin{bmatrix}23&11\\0&13\end{bmatrix}$, $\begin{bmatrix}25&38\\0&47\end{bmatrix}$, $\begin{bmatrix}43&47\\0&13\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.768.17-48.bld.2.1, 48.768.17-48.bld.2.2, 48.768.17-48.bld.2.3, 48.768.17-48.bld.2.4, 48.768.17-48.bld.2.5, 48.768.17-48.bld.2.6, 48.768.17-48.bld.2.7, 48.768.17-48.bld.2.8, 48.768.17-48.bld.2.9, 48.768.17-48.bld.2.10, 48.768.17-48.bld.2.11, 48.768.17-48.bld.2.12, 48.768.17-48.bld.2.13, 48.768.17-48.bld.2.14, 48.768.17-48.bld.2.15, 48.768.17-48.bld.2.16 |
Cyclic 48-isogeny field degree: | $1$ |
Cyclic 48-torsion field degree: | $16$ |
Full 48-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{82}\cdot3^{27}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{9}\cdot2^{4}$ |
Newforms: | 24.2.a.a$^{2}$, 24.2.d.a, 48.2.a.a, 96.2.d.a, 288.2.d.b$^{2}$, 576.2.a.b, 576.2.a.c$^{2}$, 576.2.a.d, 576.2.a.g, 576.2.a.h |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.192.7.em.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{6}\cdot2^{2}$ |
48.96.1.ds.1 | $48$ | $4$ | $4$ | $1$ | $1$ | $1^{8}\cdot2^{4}$ |
48.192.7.eg.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}\cdot2^{2}$ |
48.192.7.hr.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
48.192.7.hu.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{6}\cdot2^{2}$ |
48.192.9.bfm.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
48.192.9.bfn.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
48.192.9.bgk.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
48.768.33.tl.1 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tl.4 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tn.1 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tn.4 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tt.2 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tt.3 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tv.2 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.768.33.tv.3 | $48$ | $2$ | $2$ | $33$ | $3$ | $2^{6}\cdot4$ |
48.1152.65.bta.1 | $48$ | $3$ | $3$ | $65$ | $7$ | $1^{24}\cdot2^{12}$ |