Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $4$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $1^{4}\cdot2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AP7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.7.291 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&6\\0&41\end{bmatrix}$, $\begin{bmatrix}7&11\\0&43\end{bmatrix}$, $\begin{bmatrix}29&39\\0&29\end{bmatrix}$, $\begin{bmatrix}31&29\\0&5\end{bmatrix}$, $\begin{bmatrix}43&8\\0&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.7.hu.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $1$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{26}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a$^{2}$, 24.2.d.a, 48.2.a.a, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w - z^{2} $ |
| $=$ | $y^{2} - t^{2} - u v$ | |
| $=$ | $y w + z u - w t$ | |
| $=$ | $x y + x t + z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} y^{2} + 2 x^{6} z^{2} - 2 x^{4} y^{4} + 7 x^{4} y^{2} z^{2} - 2 x^{4} z^{4} + 2 x^{2} y^{4} z^{2} + \cdots + y^{2} z^{6} $ |
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.
| Canonical model |
|---|
| $(0:1:0:0:0:1:1)$, $(0:-1:0:0:0:1:1)$, $(0:0:0:0:1:-1:1)$, $(0:0:0:0:-1:-1:1)$ |
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.gi.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z-t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-2X^{2}Y^{2}+XY^{3}+X^{3}Z+X^{2}YZ-3XY^{2}Z+X^{2}Z^{2}+Y^{2}Z^{2}-YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.192.7.hu.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{6}Y^{2}+2X^{6}Z^{2}-2X^{4}Y^{4}+7X^{4}Y^{2}Z^{2}-2X^{4}Z^{4}+2X^{2}Y^{4}Z^{2}+X^{2}Y^{2}Z^{4}+Y^{2}Z^{6} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 16.96.0-16.x.2.2 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.0-16.x.2.2 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.gi.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-24.gi.2.24 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qd.1.7 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qd.1.33 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qj.1.50 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.qj.1.61 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
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.oa.3.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.oa.4.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.oi.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.oi.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pf.3.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pf.4.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pn.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pn.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.17-48.is.2.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.jo.2.8 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.pv.2.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.qa.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bkw.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bld.2.7 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bme.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bmj.2.7 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bqj.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bqj.3.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bqr.3.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bqr.4.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.brq.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.brq.3.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bry.3.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bry.4.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.1152.29-48.nm.1.18 | $48$ | $3$ | $3$ | $29$ | $0$ | $1^{10}\cdot2^{6}$ |