Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 48.48.1.266 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&5\\44&13\end{bmatrix}$, $\begin{bmatrix}17&7\\32&23\end{bmatrix}$, $\begin{bmatrix}31&29\\36&31\end{bmatrix}$, $\begin{bmatrix}35&11\\8&9\end{bmatrix}$, $\begin{bmatrix}35&34\\36&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.24.1.a.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $24576$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 36x $ |
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
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 |
|---|---|---|---|---|---|---|
| 16.24.0-8.n.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-8.n.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.a.2.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.d.1.13 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.g.1.19 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.i.1.13 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bo.1.11 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bo.2.15 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bp.1.14 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bp.2.6 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bq.1.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bq.2.11 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.br.1.16 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.br.2.12 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bs.1.14 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bs.2.16 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bt.1.11 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bt.2.9 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bu.1.10 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bu.2.14 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bv.1.15 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.bv.2.11 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.cf.1.7 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.cg.1.1 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.cj.1.7 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-48.ck.1.5 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.144.5-48.e.1.52 | $48$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5-48.op.1.55 | $48$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
| 240.96.1-240.cg.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ch.1.21 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ck.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cl.1.13 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cw.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cw.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cx.1.31 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cx.2.26 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cy.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cy.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cz.1.31 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.cz.2.26 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.da.1.29 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.da.2.30 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.db.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.db.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dc.1.29 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dc.2.30 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dd.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.dd.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ec.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.ed.1.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.eg.1.15 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.eh.1.11 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.240.9-240.a.1.52 | $240$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 240.288.9-240.mi.1.65 | $240$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 240.480.17-240.fq.1.54 | $240$ | $10$ | $10$ | $17$ | $?$ | not computed |