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: | 48.288.10.178 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}15&38\\8&9\end{bmatrix}$, $\begin{bmatrix}17&16\\40&1\end{bmatrix}$, $\begin{bmatrix}27&38\\8&33\end{bmatrix}$, $\begin{bmatrix}33&14\\32&33\end{bmatrix}$, $\begin{bmatrix}35&20\\8&37\end{bmatrix}$, $\begin{bmatrix}41&22\\40&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.10.p.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: | $2^{46}\cdot3^{20}$ |
| 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 $ | $=$ | $ t r + u v $ |
| $=$ | $y s - w r$ | |
| $=$ | $x s + w u$ | |
| $=$ | $x r + y u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{13} - 8 x^{9} y^{4} - 36 x^{6} y^{5} z^{2} + 5 x^{5} y^{8} + 36 x^{2} y^{9} z^{2} + \cdots + 432 y^{7} z^{6} $ |
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 |
|---|
| $(0:0:0:0:-1:0:0:0:1:0)$, $(0:0:0:0:0:0:-1/2:1/2:1:1)$, $(0:0:0:0:0:1:0:0:0:0)$, $(0:0:0:0:0:0:1/2:1/2:-1:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -z$ |
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.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{12}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{13}-8X^{9}Y^{4}-36X^{6}Y^{5}Z^{2}+5X^{5}Y^{8}+36X^{2}Y^{9}Z^{2}-XY^{12}+432Y^{7}Z^{6} $ |
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.1.15 | $48$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
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.2.9 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ij.2.35 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kc.2.27 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ks.1.18 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kt.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nm.2.29 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nn.2.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nq.2.19 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.nt.2.22 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.nu.2.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.nv.2.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.pm.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pn.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pq.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ps.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pt.2.2 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pu.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pw.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.px.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.py.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pz.2.2 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qa.1.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qb.2.1 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qd.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qe.1.11 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qf.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qh.2.2 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qi.1.3 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qj.2.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qk.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ql.2.1 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qm.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qn.1.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qo.2.1 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.qp.1.1 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rb.1.4 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rc.2.4 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ro.2.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rp.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |