Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $1^{2}\cdot2^{7}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AN9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.9.8907 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&3\\0&43\end{bmatrix}$, $\begin{bmatrix}19&23\\12&31\end{bmatrix}$, $\begin{bmatrix}25&24\\24&41\end{bmatrix}$, $\begin{bmatrix}43&20\\12&25\end{bmatrix}$, $\begin{bmatrix}47&46\\12&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.9.bhd.2 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: | $2^{39}\cdot3^{9}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}\cdot4$ |
| Newforms: | 24.2.a.a, 24.2.f.a, 96.2.c.a, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x z + y t $ |
| $=$ | $x s - w t + t r$ | |
| $=$ | $w t + t v + u s$ | |
| $=$ | $y v + w v + w r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 12 x^{6} y^{2} z^{4} - 6 x^{6} z^{6} - 16 x^{4} y^{8} + 14 x^{4} y^{4} z^{4} + x^{4} y^{2} z^{6} + \cdots + y^{8} z^{4} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:2:0:1:0:1:0:0:0)$, $(-1:-2:0:-1:0:1:0:0:0)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x-u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x-u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t$ |
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.bhd.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -12X^{6}Y^{2}Z^{4}-6X^{6}Z^{6}-16X^{4}Y^{8}+14X^{4}Y^{4}Z^{4}+X^{4}Y^{2}Z^{6}-4X^{2}Y^{10}-18X^{2}Y^{8}Z^{2}-8X^{2}Y^{6}Z^{4}+2Y^{12}+3Y^{10}Z^{2}+Y^{8}Z^{4} $ |
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 | $24$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-24.gf.2.5 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-48.pz.3.15 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.3-48.pz.3.32 | $48$ | $2$ | $2$ | $3$ | $0$ | $2\cdot4$ |
| 48.192.5-48.ou.2.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.192.5-48.ou.2.18 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.768.17-48.gp.2.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.kv.3.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.nz.2.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.ok.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bbr.3.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bby.4.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bci.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bcn.2.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bdz.8.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bep.6.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bet.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bez.4.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bin.2.9 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.biq.4.11 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bje.2.15 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bjf.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.768.17-48.bob.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.boe.2.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bos.4.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bot.2.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bql.2.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bqs.2.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brc.4.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.brh.3.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.1152.33-48.cit.1.23 | $48$ | $3$ | $3$ | $33$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{3}$ |