Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| 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 | $2^{4}$ | ||
| 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: | 48.96.1.1563 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&27\\28&41\end{bmatrix}$, $\begin{bmatrix}23&16\\28&33\end{bmatrix}$, $\begin{bmatrix}23&36\\4&29\end{bmatrix}$, $\begin{bmatrix}35&43\\16&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.t.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $12288$ |
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 $ | $=$ | $ 6 x z - w^{2} $ |
| $=$ | $96 x^{2} + y^{2} - 6 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 x^{2} y^{2} - 36 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{y^{12}-672y^{8}w^{4}+162048y^{4}w^{8}+191056320z^{12}-253808640z^{8}w^{4}+112803840z^{4}w^{8}-16769024w^{12}}{w^{4}(y^{8}+48y^{4}w^{4}-1296z^{8}+576z^{4}w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.t.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-6X^{2}Y^{2}-36Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-16.b.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.bj.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.h.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.h.1.19 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-24.bj.1.6 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-16.b.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.192.1-48.cr.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cr.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cs.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cs.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ct.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.ct.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cu.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cu.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.cp.1.5 | $48$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 48.384.9-48.zs.1.1 | $48$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
| 96.192.5-96.s.1.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.s.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bi.1.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bi.2.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bk.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.bk.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cg.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cg.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.gv.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gv.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gw.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gw.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gx.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gx.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gy.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.gy.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.480.17-240.bj.1.1 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |