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: | 48.288.10.9 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&14\\8&21\end{bmatrix}$, $\begin{bmatrix}9&28\\40&45\end{bmatrix}$, $\begin{bmatrix}15&20\\8&9\end{bmatrix}$, $\begin{bmatrix}17&30\\24&1\end{bmatrix}$, $\begin{bmatrix}29&24\\24&5\end{bmatrix}$, $\begin{bmatrix}47&22\\16&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.10.j.2 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 u - v r $ |
| $=$ | $x u - z t$ | |
| $=$ | $w s - u r$ | |
| $=$ | $w a - t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} y^{4} - 4 x^{4} z^{5} + 4 y^{8} z $ |
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 |
|---|
| $(0:0:0:0:0:0:0:0:1:0)$, $(0:0:0:0:-1/2:1/2:0:0:1:1)$, $(0:0:0:0:0:0:0:0:0:1)$, $(0:0:0:0:1/2:1/2:0:0:-1:1)$ |
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 24.72.4.ch.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -v$ |
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.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}Y^{4}-4X^{4}Z^{5}+4Y^{8}Z $ |
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.1.10 | $16$ | $3$ | $3$ | $2$ | $0$ | $1^{4}\cdot4$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.2-16.d.1.10 | $16$ | $3$ | $3$ | $2$ | $0$ | $1^{4}\cdot4$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
| 48.144.4-24.ch.1.12 | $48$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
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.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hp.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hu.1.30 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hv.1.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ic.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iw.2.33 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iy.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.iz.2.26 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ja.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jb.2.18 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jc.2.13 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jd.1.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jf.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jg.2.15 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jh.2.14 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ji.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jj.2.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jk.1.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jl.2.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jm.2.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jn.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jo.2.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jp.2.6 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jq.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.js.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jt.1.7 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ju.2.13 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jv.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jx.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jy.2.26 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jz.2.14 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ka.2.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kb.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kf.1.21 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kg.2.9 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kh.1.18 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kq.1.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kr.1.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ks.1.18 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kt.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{3}\cdot2\cdot4$ |