Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $2^{4}$ | ||
| 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: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48B9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.9.71 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}9&28\\4&27\end{bmatrix}$, $\begin{bmatrix}13&28\\16&41\end{bmatrix}$, $\begin{bmatrix}19&41\\40&25\end{bmatrix}$, $\begin{bmatrix}27&29\\20&15\end{bmatrix}$, $\begin{bmatrix}47&5\\40&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.9.ew.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{42}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{2}$, 64.2.a.a, 288.2.d.a$^{2}$, 576.2.a.a, 576.2.a.i |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ u s + v r $ |
| $=$ | $y z + t s$ | |
| $=$ | $x y + t v$ | |
| $=$ | $x y + w s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} z^{2} + x^{4} z^{4} + 2 x^{2} y^{6} + 2 y^{6} z^{2} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31,127$, 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 24.72.4.ge.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
| $\displaystyle W$ | $=$ | $\displaystyle s$ |
Equation of the image curve:
| $0$ | $=$ | $ 2XY-ZW $ |
| $=$ | $ 16X^{3}-2Y^{3}+4XZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.144.9.ew.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}Z^{2}+X^{4}Z^{4}+2X^{2}Y^{6}+2Y^{6}Z^{2} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 16.96.1-16.s.2.7 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.s.2.7 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.144.4-24.ge.2.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 48.144.4-48.bf.2.7 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 48.144.4-48.bf.2.50 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 48.144.4-24.ge.2.3 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 48.144.5-48.a.1.59 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.144.5-48.a.1.62 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.17-48.ee.1.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.fr.1.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.oa.2.11 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.oo.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.baa.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bai.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bbg.2.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bbo.1.4 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bey.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bfg.1.16 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bge.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bgm.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bho.1.23 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bhw.1.15 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.biy.2.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bjg.2.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bke.2.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bkm.2.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.blk.1.16 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bls.1.14 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bmu.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bnc.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bob.2.11 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.576.17-48.bok.2.11 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |