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: | $20$ (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: | 48.384.7.170 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&24\\12&19\end{bmatrix}$, $\begin{bmatrix}7&39\\0&5\end{bmatrix}$, $\begin{bmatrix}11&41\\24&13\end{bmatrix}$, $\begin{bmatrix}19&1\\12&19\end{bmatrix}$, $\begin{bmatrix}19&9\\12&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.7.hw.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $8$ |
| Full 48-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{27}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 24.2.d.a$^{2}$, 192.2.a.b, 192.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ z u + w v $ |
| $=$ | $x t + z t - w v$ | |
| $=$ | $x y - y w + z t + w t$ | |
| $=$ | $y t - y v - 2 z^{2} - t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y^{3} - 2 x^{6} y^{2} z + x^{6} y z^{2} + 2 x^{4} y^{4} z - 6 x^{4} y^{3} z^{2} - 6 x^{4} y z^{4} + \cdots - 8 y z^{8} $ |
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 |
|---|
| $(0:1:0:0:1:0:0)$, $(0:1:0:0:0:0:0)$, $(0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:1)$ |
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.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
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.hw.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}Y^{3}-2X^{6}Y^{2}Z+2X^{4}Y^{4}Z+X^{6}YZ^{2}-6X^{4}Y^{3}Z^{2}-6X^{4}YZ^{4}-4X^{2}Y^{3}Z^{4}+2X^{4}Z^{5}-24X^{2}Y^{2}Z^{5}-4X^{2}YZ^{6}-8Y^{3}Z^{6}-16Y^{2}Z^{7}-8YZ^{8} $ |
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.z.1.2 | $16$ | $4$ | $4$ | $0$ | $0$ | 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.z.1.2 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.gi.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-24.gi.1.12 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qa.2.13 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qa.2.15 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qk.1.4 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.qk.1.57 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
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.ok.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.ok.4.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.os.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.os.2.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pp.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pp.4.6 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.px.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.px.2.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.17-48.ih.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.lg.2.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.oq.2.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.qv.1.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.blg.1.2 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.blp.2.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bmm.2.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bmt.2.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bqt.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bqt.2.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.brb.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.brb.4.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bsa.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bsa.2.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bsi.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bsi.4.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.1152.29-48.nw.1.8 | $48$ | $3$ | $3$ | $29$ | $2$ | $1^{10}\cdot2^{6}$ |