Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $3^{4}\cdot6^{2}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48E8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.8.142 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&42\\12&25\end{bmatrix}$, $\begin{bmatrix}17&25\\16&31\end{bmatrix}$, $\begin{bmatrix}25&45\\0&31\end{bmatrix}$, $\begin{bmatrix}31&37\\44&31\end{bmatrix}$, $\begin{bmatrix}43&43\\20&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.8.ji.2 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: | $2^{32}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{2}$, 72.2.d.a, 288.2.d.a, 576.2.a.e, 576.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations
| $ 0 $ | $=$ | $ x u - t r $ |
| $=$ | $w u + t v$ | |
| $=$ | $x v + w r$ | |
| $=$ | $x u - x r + z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 16 x^{10} - 160 x^{9} y - 800 x^{8} y^{2} - 2560 x^{7} y^{3} - 5624 x^{6} y^{4} - 8656 x^{5} y^{5} + \cdots + 16 y^{4} z^{6} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
| $\displaystyle W$ | $=$ | $\displaystyle -r$ |
Equation of the image curve:
| $0$ | $=$ | $ XZ+YW $ |
| $=$ | $ 8X^{3}+4XY^{2}-Z^{3}-ZW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.144.8.ji.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ -16X^{10}-160X^{9}Y-800X^{8}Y^{2}-2560X^{7}Y^{3}-5624X^{6}Y^{4}-8656X^{5}Y^{5}-1312X^{5}Y^{2}Z^{3}-9264X^{4}Y^{6}-6560X^{4}Y^{3}Z^{3}-6656X^{3}Y^{7}-14080X^{3}Y^{4}Z^{3}-2977X^{2}Y^{8}-16000X^{2}Y^{5}Z^{3}-706XY^{9}-9592XY^{6}Z^{3}-66Y^{10}-2424Y^{7}Z^{3}+16Y^{4}Z^{6} $ |
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 |
| 16.96.0-16.y.1.1 | $16$ | $3$ | $3$ | $0$ | $0$ | 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.y.1.1 | $16$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.144.4-24.ge.1.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.y.2.5 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.y.2.15 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 48.144.4-48.bl.1.7 | $48$ | $2$ | $2$ | $4$ | $1$ | $2^{2}$ |
| 48.144.4-48.bl.1.35 | $48$ | $2$ | $2$ | $4$ | $1$ | $2^{2}$ |
| 48.144.4-24.ge.1.30 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.15-48.ye.2.6 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.ym.2.1 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.zk.2.6 | $48$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.zs.2.1 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bbk.2.5 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bbw.2.2 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bcu.1.3 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.15-48.bdc.1.2 | $48$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 48.576.17-48.ee.2.1 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.jk.2.12 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.lw.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.qs.2.7 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cki.2.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ckq.2.3 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.clo.2.2 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.clw.2.2 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cwa.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cwi.1.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cxg.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cxo.2.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cyq.1.7 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cyy.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.czz.1.7 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dah.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |