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 | $2^{6}\cdot4$ | ||
| 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: | 48.192.1.101 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&20\\40&7\end{bmatrix}$, $\begin{bmatrix}7&22\\16&11\end{bmatrix}$, $\begin{bmatrix}23&26\\40&35\end{bmatrix}$, $\begin{bmatrix}35&38\\36&29\end{bmatrix}$, $\begin{bmatrix}41&2\\4&27\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.1.i.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| 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 $ | $=$ | $ 2 y^{2} - z^{2} - w^{2} $ |
| $=$ | $3 x^{2} + z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 2 y^{2} z^{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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{8}-z^{4}w^{4}+w^{8})^{3}}{w^{8}z^{8}(z-w)^{2}(z+w)^{2}(z^{2}+w^{2})^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.i.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-2Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.l.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.d.2.19 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.d.2.20 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.l.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bf.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bf.2.15 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bh.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-48.bh.2.10 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-48.b.2.18 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.b.2.19 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bz.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bz.2.10 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cb.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cb.2.15 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.bp.1.12 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.bq.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.ck.1.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.cl.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.er.1.11 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.es.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.ev.1.12 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.ew.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.ez.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fa.1.12 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fd.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fe.1.12 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.576.17-48.ea.2.20 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.hx.2.20 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 240.384.5-240.bep.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.beq.1.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bff.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bfg.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.blr.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bls.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bmd.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bme.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bmh.1.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bmi.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bmt.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bmu.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |