Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{8}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AO7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.7.2130 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&0\\0&37\end{bmatrix}$, $\begin{bmatrix}7&23\\36&43\end{bmatrix}$, $\begin{bmatrix}19&32\\12&17\end{bmatrix}$, $\begin{bmatrix}23&10\\12&37\end{bmatrix}$, $\begin{bmatrix}29&5\\0&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.7.et.3 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{31}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{3}$ |
| Newforms: | 24.2.a.a, 24.2.f.a, 96.2.d.a, 192.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y u + w v $ |
| $=$ | $x u - y z - y w + y t - u^{2} - u v$ | |
| $=$ | $x u - y^{2} + y z + y t + u^{2} - u v$ | |
| $=$ | $x y + y u + y v - 2 t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{6} y^{2} + 12 x^{6} z^{2} - 2 x^{4} y^{4} - 10 x^{4} y^{2} z^{2} + 8 x^{4} z^{4} + \cdots + 8 y^{2} z^{6} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.gf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x+v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x+v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-XY^{3}-X^{2}Z^{2}-Y^{2}Z^{2}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.7.et.3 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{6}Y^{2}-2X^{4}Y^{4}+12X^{6}Z^{2}-10X^{4}Y^{2}Z^{2}+4X^{2}Y^{4}Z^{2}+8X^{4}Z^{4}+4X^{2}Y^{2}Z^{4}+8Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-24.gf.2.15 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.pw.1.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.pw.1.49 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.pz.2.8 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.pz.2.50 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.13-48.ls.3.8 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.lt.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.mh.1.3 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.mk.1.4 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.oc.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.oh.3.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.or.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.oy.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.17-48.gx.3.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.kv.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.ob.3.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.ok.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.zg.1.15 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.zl.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.zv.3.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bac.2.13 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.beb.5.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bep.3.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bev.3.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bez.4.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bgc.1.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bgd.1.14 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bgr.3.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bgu.3.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.1152.29-48.lb.1.4 | $48$ | $3$ | $3$ | $29$ | $0$ | $1^{4}\cdot2^{7}\cdot4$ |