Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| 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: | 16E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.162 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&6\\0&13\end{bmatrix}$, $\begin{bmatrix}13&38\\40&21\end{bmatrix}$, $\begin{bmatrix}39&41\\20&23\end{bmatrix}$, $\begin{bmatrix}39&46\\16&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.p.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $16$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $12288$ |
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 $ | $=$ | $ 6 x^{2} + z w $ |
| $=$ | $y^{2} - 8 z^{2} - 2 w^{2}$ |
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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{4}+4z^{2}w^{2}+w^{4})^{3}}{w^{2}z^{8}(4z^{2}+w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
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 |
|---|---|---|---|---|---|---|
| 8.48.0-8.y.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-8.y.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.j.1.11 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.j.1.16 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.d.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-48.d.1.12 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.192.3-48.fc.1.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.3-48.fd.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.3-48.hz.1.7 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.192.3-48.ia.1.5 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.192.3-48.jx.1.7 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.3-48.jy.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.3-48.kn.1.6 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.192.3-48.ko.1.5 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.288.9-48.ck.1.10 | $48$ | $3$ | $3$ | $9$ | $4$ | $1^{8}$ |
| 48.384.9-48.rs.1.8 | $48$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
| 240.192.3-240.baj.1.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bak.1.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bbx.1.15 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bby.1.7 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bhn.1.15 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bho.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.bit.1.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.biu.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.480.17-240.bf.1.8 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |