Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| 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: | 48.48.1.450 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&43\\0&47\end{bmatrix}$, $\begin{bmatrix}33&10\\8&33\end{bmatrix}$, $\begin{bmatrix}33&34\\28&31\end{bmatrix}$, $\begin{bmatrix}41&46\\32&29\end{bmatrix}$, $\begin{bmatrix}43&6\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.24.1.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: | $24576$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
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 |
|---|
| $(0:0:1)$, $(1:0:1)$, $(0:1:0)$, $(-1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{678x^{2}y^{4}z^{2}-4095x^{2}z^{6}-44xy^{6}z+4053xy^{2}z^{5}+y^{8}-4140y^{4}z^{4}+4096z^{8}}{zy^{2}(2x^{2}y^{2}z+xy^{4}+xz^{4}+y^{2}z^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.24.0-8.n.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.24.0-8.n.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 48.96.1-16.b.1.12 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.f.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.h.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.j.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.s.1.16 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.t.1.16 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.u.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.u.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.v.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.v.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.w.1.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.w.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.w.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.x.1.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.x.2.3 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.x.1.8 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bk.1.15 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bk.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bl.1.15 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bl.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bm.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bm.2.15 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bn.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-48.bn.2.15 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.144.5-48.b.1.49 | $48$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5-48.mi.1.42 | $48$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 96.96.3-32.a.1.2 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.a.2.2 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.a.1.29 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.a.2.29 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.c.1.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.c.2.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.c.1.12 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.c.2.12 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.d.1.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.d.2.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.d.1.16 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.d.2.24 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.e.1.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-32.e.2.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.e.1.12 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3-96.e.2.20 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.1-80.s.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.s.1.30 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.t.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.t.1.30 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.w.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.w.1.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.x.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.x.1.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bk.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bk.2.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bk.1.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bk.2.22 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bl.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bl.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bl.1.22 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bl.2.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bm.1.7 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bm.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bm.1.19 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bm.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bn.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-80.bn.2.3 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bn.1.10 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.96.1-240.bn.2.12 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.240.9-80.b.1.20 | $240$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 240.288.9-80.j.1.24 | $240$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 240.480.17-80.cn.1.43 | $240$ | $10$ | $10$ | $17$ | $?$ | not computed |