Invariants
| Level: | $48$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-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 | $4^{8}\cdot8^{8}$ | Cusp orbits | $4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.440 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&31\\22&23\end{bmatrix}$, $\begin{bmatrix}13&24\\16&5\end{bmatrix}$, $\begin{bmatrix}15&40\\22&33\end{bmatrix}$, $\begin{bmatrix}23&44\\0&23\end{bmatrix}$, $\begin{bmatrix}29&12\\42&43\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 48-isogeny field degree: | $32$ |
| Cyclic 48-torsion field degree: | $512$ |
| 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 $ | $=$ | $ 3 x^{2} - 6 y^{2} - 2 z^{2} - 2 w^{2} $ |
| $=$ | $6 x y - z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z - 24 x^{2} y^{2} + 2 x^{2} z^{2} - 4 x z^{3} - 72 y^{4} - 24 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(z^{2}-2zw+3w^{2})^{3}(3z^{2}+2zw+w^{2})^{3}(5z^{4}+4z^{3}w+10z^{2}w^{2}-4zw^{3}+5w^{4})^{3}}{(z^{2}+w^{2})^{4}(z^{2}+2zw-w^{2})^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1.br.1 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.192.5.gl.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5.ik.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 48.192.5.il.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 48.192.5.im.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5.io.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5.iq.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 48.192.5.is.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 48.192.5.it.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 48.192.9.sb.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 48.192.9.sf.1 | $48$ | $2$ | $2$ | $9$ | $5$ | $1^{6}\cdot2$ |
| 48.192.9.so.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 48.192.9.st.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 48.288.17.vg.1 | $48$ | $3$ | $3$ | $17$ | $8$ | $1^{8}\cdot2^{4}$ |
| 48.384.17.ur.1 | $48$ | $4$ | $4$ | $17$ | $5$ | $1^{16}$ |
| 240.192.5.cat.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cav.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cbd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cbg.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cbk.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cbn.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cbr.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.cbv.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.9.deg.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.dep.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.dfk.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.dft.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |