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 $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AO7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.384.7.2125 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&22\\24&13\end{bmatrix}$, $\begin{bmatrix}11&21\\12&7\end{bmatrix}$, $\begin{bmatrix}13&34\\0&29\end{bmatrix}$, $\begin{bmatrix}17&0\\0&29\end{bmatrix}$, $\begin{bmatrix}41&15\\24&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.192.7.fn.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: | $2^{31}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{3}$ |
| Newforms: | 24.2.a.a, 48.2.c.a, 96.2.d.a, 96.2.f.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y u - z w $ |
| $=$ | $x w - z v - w u$ | |
| $=$ | $x t - y z + z v + t u$ | |
| $=$ | $x y + 2 x v + z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{10} + x^{8} y^{2} - 6 x^{8} z^{2} - x^{6} y^{2} z^{2} + 9 x^{6} z^{4} + x^{4} y^{4} z^{2} + \cdots - 3 y^{4} z^{6} $ |
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 |
|---|
| $(-1:-2:0:0:0:0:1)$, $(1:-2:0:0:0:0:1)$ |
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.gf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x-u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x-u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
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.fn.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{10}+X^{8}Y^{2}-6X^{8}Z^{2}-X^{6}Y^{2}Z^{2}+X^{4}Y^{4}Z^{2}+9X^{6}Z^{4}-9X^{4}Y^{2}Z^{4}+2X^{2}Y^{4}Z^{4}+9X^{2}Y^{2}Z^{6}-3Y^{4}Z^{6} $ |
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 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-24.gf.2.28 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.pv.2.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.pv.2.16 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.qe.3.39 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 48.192.3-48.qe.3.49 | $48$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
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.mp.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.ms.2.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.ng.4.6 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.nh.2.8 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.oz.2.3 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.pg.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.pq.4.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.13-48.pv.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{2}$ |
| 48.768.17-48.gp.2.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.kh.4.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.rb.3.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.ri.1.4 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bad.3.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bak.4.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bau.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.baz.2.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bdz.3.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bej.7.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bfj.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bfn.3.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{2}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bgz.2.9 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bhc.4.11 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bhq.2.15 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.768.17-48.bhr.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{3}$ |
| 48.1152.29-48.kh.2.6 | $48$ | $3$ | $3$ | $29$ | $0$ | $1^{4}\cdot2^{7}\cdot4$ |