Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $10 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $12^{4}\cdot48^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48A10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.10.10 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&34\\8&37\end{bmatrix}$, $\begin{bmatrix}1&44\\32&17\end{bmatrix}$, $\begin{bmatrix}5&42\\0&37\end{bmatrix}$, $\begin{bmatrix}23&32\\16&41\end{bmatrix}$, $\begin{bmatrix}25&14\\40&1\end{bmatrix}$, $\begin{bmatrix}47&12\\0&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.10.j.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: | $2^{46}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2\cdot4$ |
| Newforms: | 36.2.a.a$^{3}$, 64.2.b.a, 144.2.a.a, 576.2.d.c |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ x r - a^{2} $ |
| $=$ | $y v - s^{2}$ | |
| $=$ | $x^{2} + z a$ | |
| $=$ | $y^{2} - z s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{8} z - 2 x^{4} y^{5} + y^{4} z^{5} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:0:-1:-1:1:0:0)$, $(0:0:0:0:0:0:1:0:0:0)$, $(0:0:0:0:1:0:1:1:0:0)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -a$ |
| $\displaystyle W$ | $=$ | $\displaystyle -s$ |
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 48.144.10.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{8}Z-2X^{4}Y^{5}+Y^{4}Z^{5} $ |
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 |
| 16.96.2-16.d.2.9 | $16$ | $3$ | $3$ | $2$ | $0$ | $1^{4}\cdot4$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.2-16.d.2.9 | $16$ | $3$ | $3$ | $2$ | $0$ | $1^{4}\cdot4$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
| 48.144.4-24.ch.1.4 | $48$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.19-48.ho.1.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hp.2.9 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hu.2.30 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hv.2.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ic.2.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iw.1.41 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iy.2.14 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iz.1.24 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ja.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jb.1.25 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jc.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jd.2.2 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jf.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jg.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jh.1.14 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ji.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jj.1.6 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jk.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jl.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jm.1.6 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jn.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jo.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jp.1.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jq.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.js.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jt.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ju.1.8 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jv.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jx.2.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jy.1.26 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jz.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ka.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kb.2.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kf.2.25 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kg.1.7 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kh.2.17 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kq.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kr.2.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ks.2.17 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kt.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{3}\cdot2\cdot4$ |