Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AO9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.9.8900 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&21\\24&47\end{bmatrix}$, $\begin{bmatrix}7&35\\12&23\end{bmatrix}$, $\begin{bmatrix}11&13\\12&7\end{bmatrix}$, $\begin{bmatrix}37&34\\0&1\end{bmatrix}$, $\begin{bmatrix}41&36\\24&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.9.bft.1 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: | $2^{43}\cdot3^{15}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 96.2.d.a, 288.2.a.b, 288.2.a.c, 288.2.a.d$^{2}$, 288.2.d.b |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x r + v s $ |
| $=$ | $x s + y w - z w + w u$ | |
| $=$ | $x r - y w + z w - w t$ | |
| $=$ | $y t - z t + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} y^{2} - 36 x^{6} y^{4} + 16 x^{6} y^{2} z^{2} - 12 x^{6} z^{4} + 18 x^{4} y^{6} + \cdots + y^{2} z^{8} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no 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 :
| $\displaystyle X$ | $=$ | $\displaystyle -x-v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x-v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
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 48.192.9.bft.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{8}Y^{2}-36X^{6}Y^{4}+16X^{6}Y^{2}Z^{2}-12X^{6}Z^{4}+18X^{4}Y^{6}-30X^{4}Y^{4}Z^{2}+14X^{4}Y^{2}Z^{4}-18X^{4}Z^{6}+9X^{2}Y^{6}Z^{2}-6X^{2}Y^{4}Z^{4}+11X^{2}Y^{2}Z^{6}-6X^{2}Z^{8}+Y^{2}Z^{8} $ |
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 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.96.1-48.ca.1.10 | $48$ | $4$ | $4$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.192.3-24.gf.2.29 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.3-48.qc.2.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.3-48.qc.2.38 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 48.192.5-48.oq.1.8 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.192.5-48.oq.1.45 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.17-48.hv.2.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.le.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.po.1.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.qo.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bgm.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bgm.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bgu.3.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bgu.4.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bhr.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bhr.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bhz.1.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bhz.2.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.768.17-48.bix.1.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bix.2.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bjf.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bjf.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bke.3.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bke.4.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bkm.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bkm.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{2}\cdot4$ |
| 48.768.17-48.bkp.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bls.2.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bmd.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bms.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.1152.33-48.cjp.2.21 | $48$ | $3$ | $3$ | $33$ | $3$ | $1^{12}\cdot2^{6}$ |