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: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4\cdot16$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.48.1.27 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&44\\24&29\end{bmatrix}$, $\begin{bmatrix}23&35\\20&13\end{bmatrix}$, $\begin{bmatrix}25&46\\8&13\end{bmatrix}$, $\begin{bmatrix}35&14\\40&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.24.1.d.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $16$ |
| Cyclic 48-torsion field degree: | $256$ |
| 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} + 36x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^8}\cdot\frac{243x^{2}y^{4}z^{2}-36xy^{6}z+19683xy^{2}z^{5}+y^{8}+531441z^{8}}{z^{5}y^{2}x}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.o.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.24.0-8.o.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ | 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.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.e.1.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.n.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.p.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cm.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cp.1.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.cq.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.ct.1.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.144.5-48.h.1.19 | $48$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5-48.os.1.10 | $48$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 240.96.1-240.cq.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cr.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cu.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cv.1.23 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.em.1.9 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.en.1.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.eq.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.er.1.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.240.9-240.d.1.18 | $240$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 240.288.9-240.ml.1.41 | $240$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 240.480.17-240.ft.1.22 | $240$ | $10$ | $10$ | $17$ | $?$ | not computed |