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: | $8$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48A9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.9.10 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&14\\20&45\end{bmatrix}$, $\begin{bmatrix}5&28\\32&41\end{bmatrix}$, $\begin{bmatrix}7&40\\44&1\end{bmatrix}$, $\begin{bmatrix}33&20\\32&21\end{bmatrix}$, $\begin{bmatrix}35&26\\4&29\end{bmatrix}$, $\begin{bmatrix}41&0\\0&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.9.a.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{40}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 64.2.a.a, 144.2.a.a, 576.2.a.a, 576.2.a.e, 576.2.a.f, 576.2.a.i |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ w r + t u $ |
| $=$ | $x y + u s - v s$ | |
| $=$ | $z u + u r + v r$ | |
| $=$ | $x s - w r + t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{8} - x^{4} z^{4} + 4 y^{6} z^{2} $ |
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 |
|---|
| $(0:0:1:0:0:0:0:0:0)$, $(0:0:-1:0:0:0:0:1:0)$ |
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.ch.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
| $\displaystyle W$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 4Y^{2}+ZW $ |
| $=$ | $ X^{3}+YZ^{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.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{8}-X^{4}Z^{4}+4Y^{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.a.1.4 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{8}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.1-16.a.1.4 | $16$ | $3$ | $3$ | $1$ | $0$ | $1^{8}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.4-48.bl.1.30 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 48.144.4-48.bl.1.35 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 48.144.4-24.ch.1.30 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 48.144.5-48.a.1.30 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 48.144.5-48.a.1.35 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
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.c.1.13 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.d.1.10 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 48.576.17-48.k.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.k.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.m.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.m.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.bi.1.10 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bj.1.18 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 48.576.17-48.bq.1.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.br.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.ca.1.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.ca.2.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.cc.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.cc.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.cy.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 48.576.17-48.cz.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 48.576.17-48.dg.1.20 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.dg.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.di.1.18 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.di.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.ee.1.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.ee.2.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.eg.1.11 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.17-48.eg.2.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{4}$ |
| 48.576.19-48.iv.2.35 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.ix.1.28 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.iy.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.iy.2.14 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.lx.1.12 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.ly.1.12 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.lz.1.13 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.lz.2.9 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.of.1.15 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.of.2.15 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.og.2.12 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.oh.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{10}$ |
| 48.576.19-48.pn.1.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.pn.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $2^{3}\cdot4$ |
| 48.576.19-48.po.2.2 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.pp.2.2 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |