Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.1.2587 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&19\\0&25\end{bmatrix}$, $\begin{bmatrix}17&32\\32&25\end{bmatrix}$, $\begin{bmatrix}17&38\\36&35\end{bmatrix}$, $\begin{bmatrix}45&29\\44&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.i.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - y^{2} - w^{2} $ |
| $=$ | $2 x^{2} + 3 y^{2} - z^{2} + w^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2\,\frac{(z^{4}-8z^{3}w-28z^{2}w^{2}-16zw^{3}+4w^{4})^{3}(z^{4}+8z^{3}w-28z^{2}w^{2}+16zw^{3}+4w^{4})^{3}}{w^{2}z^{2}(z^{2}-2w^{2})^{2}(z^{2}+2w^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-8.m.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.e.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.e.1.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.m.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.y.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.y.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.z.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.z.2.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-16.f.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.f.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.u.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.u.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.v.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.v.2.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-16.bn.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.bo.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fx.1.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fy.1.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.576.17-48.jk.2.10 | $48$ | $3$ | $3$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.lg.1.6 | $48$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 96.384.5-32.l.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.m.2.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.z.1.7 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.ba.2.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.9-32.r.2.3 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-32.t.1.1 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.cb.2.6 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.cf.1.4 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-80.kr.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.ks.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bpb.1.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bpc.1.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |