Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $288$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2^{2}\cdot4\cdot16$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.1.249 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&47\\28&45\end{bmatrix}$, $\begin{bmatrix}5&33\\28&5\end{bmatrix}$, $\begin{bmatrix}11&32\\36&13\end{bmatrix}$, $\begin{bmatrix}13&19\\32&39\end{bmatrix}$, $\begin{bmatrix}21&16\\32&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.24.1.b.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $24576$ |
Jacobian
| Conductor: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 9x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(0:0:1)$, $(3:0:1)$, $(-3:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^8}\cdot\frac{62208x^{2}y^{4}z^{2}+36864xy^{6}z+314928xy^{2}z^{5}+4096y^{8}+531441z^{8}}{z^{5}y^{2}x}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-8.n.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.96.1-48.b.2.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.f.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.h.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.j.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bw.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bw.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bx.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bx.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.by.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.by.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bz.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bz.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.ca.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.ca.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cb.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cb.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cc.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cc.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cd.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cd.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.ce.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.ch.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.ci.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cl.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.144.5-48.f.1.49 | $48$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5-48.oq.1.2 | $48$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 240.96.1-240.ci.1.18 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cj.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cm.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cn.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.de.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.de.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.df.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.df.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dg.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dg.2.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dh.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dh.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.di.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.di.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dj.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dj.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dk.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dk.2.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dl.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dl.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ee.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ef.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ei.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ej.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.240.9-240.b.1.6 | $240$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 240.288.9-240.mj.1.2 | $240$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 240.480.17-240.fr.1.68 | $240$ | $10$ | $10$ | $17$ | $?$ | not computed |