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: | 48.288.10.175 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&6\\0&13\end{bmatrix}$, $\begin{bmatrix}9&40\\8&21\end{bmatrix}$, $\begin{bmatrix}13&32\\32&29\end{bmatrix}$, $\begin{bmatrix}35&18\\24&13\end{bmatrix}$, $\begin{bmatrix}41&38\\8&37\end{bmatrix}$, $\begin{bmatrix}45&16\\8&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.144.10.n.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: | $2^{46}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2\cdot4$ |
| Newforms: | 36.2.a.a$^{3}$, 64.2.b.a, 144.2.a.a, 192.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ w a - u v $ |
| $=$ | $y t - z u$ | |
| $=$ | $x t + z w$ | |
| $=$ | $x a + y v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{12} + 8 x^{10} z^{2} - 24 x^{8} z^{4} + 4 x^{7} y^{4} z + 32 x^{6} z^{6} - 4 x^{5} y^{4} z^{3} + \cdots + 16 y^{8} z^{4} $ |
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 |
|---|
| $(0:1:0:0:0:0:0:0:0:0)$, $(0:0:0:0:1/2:-1/2:0:0:-1:1)$, $(0:0:1:0:0:0:0:0:0:1)$, $(0:0:0:0:1/2:1/2:0:0:1:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -v$ |
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.n.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ -X^{12}+8X^{10}Z^{2}-24X^{8}Z^{4}+4X^{7}Y^{4}Z+32X^{6}Z^{6}-4X^{5}Y^{4}Z^{3}-16X^{4}Z^{8}+64X^{3}Y^{4}Z^{5}+16Y^{8}Z^{4} $ |
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.1.31 | $48$ | $2$ | $2$ | $4$ | $0$ | $2\cdot4$ |
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.hu.1.30 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.hv.2.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.in.2.35 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kd.1.22 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.kq.1.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.kr.2.10 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.lk.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.ll.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.lq.2.20 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.lt.2.28 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.lu.1.17 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.lv.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{3}\cdot2\cdot4$ |
| 48.576.19-48.oe.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.of.2.15 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.oi.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ok.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ol.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.om.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.oo.2.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.op.2.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.oq.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.or.1.6 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.os.2.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ot.1.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ov.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ow.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ox.1.28 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.oy.1.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.oz.2.13 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pa.1.5 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pc.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pd.2.13 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pe.2.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pf.2.11 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.pg.1.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ph.2.11 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ra.2.12 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rd.2.11 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ri.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.rj.2.6 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2^{2}$ |