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 | $2^{6}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AP7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.7.213 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&43\\24&11\end{bmatrix}$, $\begin{bmatrix}23&14\\12&37\end{bmatrix}$, $\begin{bmatrix}35&0\\36&41\end{bmatrix}$, $\begin{bmatrix}41&27\\24&19\end{bmatrix}$, $\begin{bmatrix}43&37\\12&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.7.ib.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: | $2^{35}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 96.2.d.a$^{2}$, 192.2.a.b, 192.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y t - z u + z v + t u + t v $ |
| $=$ | $x z - y w + y v + z^{2} + z t$ | |
| $=$ | $x z - z^{2} - z t - w u + w v$ | |
| $=$ | $x z - y w + y u - z^{2} + z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} y^{4} + 4 x^{4} y^{2} z^{2} + x^{2} y^{6} + 14 x^{2} y^{4} z^{2} - 4 x^{2} y^{2} z^{4} + \cdots + 8 y^{4} z^{4} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=7,47$, 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 :
| $\displaystyle X$ | $=$ | $\displaystyle x+t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x+t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
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.ib.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}Y^{4}+4X^{4}Y^{2}Z^{2}+X^{2}Y^{6}+14X^{2}Y^{4}Z^{2}-4X^{2}Y^{2}Z^{4}+8X^{2}Z^{6}+4Y^{6}Z^{2}+8Y^{4}Z^{4} $ |
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.ba.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.ba.1.2 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-24.gf.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.1.37 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 48.192.3-48.qk.1.36 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 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.oq.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.oq.4.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.oy.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.oy.2.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pv.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.pv.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.qd.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.13-48.qd.2.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 48.768.17-48.if.2.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.ll.1.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.pq.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.qu.1.8 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bll.2.2 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bls.2.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bmp.1.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bmu.2.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bqz.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bqz.2.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.brh.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.brh.3.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bsg.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bsg.2.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bso.2.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.bso.4.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.1152.29-48.ob.2.8 | $48$ | $3$ | $3$ | $29$ | $2$ | $1^{10}\cdot2^{6}$ |
