Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| 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 | $2^{4}\cdot4^{2}$ | ||
| 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: | 48.192.1.330 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&22\\44&33\end{bmatrix}$, $\begin{bmatrix}21&10\\40&9\end{bmatrix}$, $\begin{bmatrix}45&22\\8&45\end{bmatrix}$, $\begin{bmatrix}47&2\\20&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.1.p.1 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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y + y^{2} - z^{2} $ |
| $=$ | $6 x^{2} + 4 x y + 4 y^{2} + 8 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 x^{2} y^{2} + z^{4} $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^8}\cdot\frac{(1296z^{8}-144z^{4}w^{4}+w^{8})^{3}}{w^{4}z^{16}(12z^{2}-w^{2})(12z^{2}+w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-6X^{2}Y^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.0-16.d.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.ba.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.d.1.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.ba.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bc.2.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bc.2.14 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bi.2.6 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bi.2.11 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-48.b.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.b.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bw.2.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bw.2.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cc.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cc.2.14 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.384.5-48.z.4.4 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.cg.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.eb.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.em.2.3 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.576.17-48.ew.2.9 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.in.1.17 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 240.384.5-240.bfx.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bgb.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bgv.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bhd.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |