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: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48B10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.10.177 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&22\\32&13\end{bmatrix}$, $\begin{bmatrix}13&24\\0&17\end{bmatrix}$, $\begin{bmatrix}19&8\\32&41\end{bmatrix}$, $\begin{bmatrix}29&14\\16&17\end{bmatrix}$, $\begin{bmatrix}43&46\\40&41\end{bmatrix}$, $\begin{bmatrix}45&46\\32&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.10.p.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: | $2^{46}\cdot3^{20}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2\cdot4$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a, 576.2.d.a, 576.2.d.b |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ x a + w u $ |
| $=$ | $x w + x s - z t$ | |
| $=$ | $2 y s - z u - u a$ | |
| $=$ | $x s - y v - y r + y a - u r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{8} z^{3} + 36 x^{7} y^{2} z^{2} + 54 x^{6} y^{4} z + 27 x^{5} y^{6} - 8 x^{4} z^{7} - 36 x^{3} y^{2} z^{6} + \cdots + 2 z^{11} $ |
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 |
|---|
| $(0:0:0:1:0:0:0:0:0:0)$, $(0:0:0:0:0:0:1:0:1:1)$, $(0:0:-1:0:0:0:0:0:0:1)$, $(0:0:0:0:0:0:0:-1:-1:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
| $\displaystyle W$ | $=$ | $\displaystyle u$ |
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 48.144.10.p.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}a$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 27X^{5}Y^{6}+54X^{6}Y^{4}Z+36X^{7}Y^{2}Z^{2}+8X^{8}Z^{3}-27XY^{6}Z^{4}-54X^{2}Y^{4}Z^{5}-36X^{3}Y^{2}Z^{6}-8X^{4}Z^{7}+2Z^{11} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
| 48.144.4-24.ch.1.7 | $48$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
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.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ij.1.42 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kc.1.22 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ks.2.17 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kt.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nm.1.29 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nn.1.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nq.1.26 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.nt.1.22 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.nu.1.3 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nv.1.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.pm.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pn.1.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pq.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ps.2.1 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pt.1.1 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pu.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pw.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.px.1.7 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.py.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pz.1.1 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qa.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qb.1.2 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qd.2.4 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qe.2.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qf.2.9 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qh.1.1 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qi.2.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qj.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qk.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ql.1.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qm.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qn.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qo.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qp.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rb.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rc.1.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ro.1.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rp.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |