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: | $8$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{4}\cdot4$ | ||
| 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: | 16E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.11 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&22\\24&5\end{bmatrix}$, $\begin{bmatrix}7&44\\32&17\end{bmatrix}$, $\begin{bmatrix}13&8\\32&9\end{bmatrix}$, $\begin{bmatrix}37&16\\24&5\end{bmatrix}$, $\begin{bmatrix}39&46\\8&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.1.b.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| 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
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 |
|---|
| $(0:1:0)$, $(-3:0:1)$, $(0:0:1)$, $(3:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^8}\cdot\frac{2025x^{2}y^{12}z^{2}+3011499x^{2}y^{8}z^{6}+645700815x^{2}y^{4}z^{10}+72xy^{14}z+452709xy^{10}z^{5}+210450636xy^{6}z^{9}+20920706406xy^{2}z^{13}+y^{16}+27702y^{12}z^{4}+17006112y^{8}z^{8}+2324522934y^{4}z^{12}+282429536481z^{16}}{z^{5}y^{4}(45x^{2}y^{4}z+26244x^{2}z^{5}+xy^{6}+8748xy^{2}z^{4}+648y^{4}z^{3})}$ |
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.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-8.i.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.i.1.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-48.i.1.9 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-48.d.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-48.d.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.h.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.h.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.q.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.q.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.3-48.bl.2.4 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.192.3-48.bs.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.bs.2.9 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.bv.1.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.bv.2.9 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.bx.1.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.3-48.cb.2.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.192.3-48.cj.1.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cj.2.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cl.1.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cl.2.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.cp.2.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.288.9-48.g.2.21 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 48.384.9-48.hr.2.10 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
| 240.192.1-240.t.1.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.t.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.bi.1.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.bi.2.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.ia.1.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ie.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ie.2.21 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ig.1.11 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ig.2.17 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ih.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.iy.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jg.1.5 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jg.2.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ji.1.9 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ji.2.17 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.jm.2.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.480.17-240.d.2.21 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |