Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.0.1532 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&32\\20&37\end{bmatrix}$, $\begin{bmatrix}21&5\\16&19\end{bmatrix}$, $\begin{bmatrix}31&4\\20&1\end{bmatrix}$, $\begin{bmatrix}47&0\\32&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.48.0.by.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $12288$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3\cdot5^{16}}\cdot\frac{(x+y)^{48}(350881x^{16}+2843136x^{15}y-94175280x^{14}y^{2}+830684160x^{13}y^{3}-901374480x^{12}y^{4}-46732142592x^{11}y^{5}+389056229568x^{10}y^{6}-1702752399360x^{9}y^{7}+5224043380320x^{8}y^{8}-10216514396160x^{7}y^{9}+14006024264448x^{6}y^{10}-10094142799872x^{5}y^{11}-1168181326080x^{4}y^{12}+6459400028160x^{3}y^{13}-4393841863680x^{2}y^{14}+795896119296xy^{15}+589345341696y^{16})^{3}}{(x+y)^{48}(x^{2}-6y^{2})^{16}(x^{2}-18xy+6y^{2})^{4}(3x^{2}-4xy+18y^{2})^{2}(23x^{4}+72x^{3}y-972x^{2}y^{2}+432xy^{3}+828y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-16.h.1.10 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 24.48.0-24.by.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-48.f.2.11 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-16.h.1.16 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 48.48.0-24.by.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 48.192.1-48.a.2.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bd.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.bl.1.9 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.cb.1.5 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.dt.2.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.ea.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.em.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.192.1-48.en.1.3 | $48$ | $2$ | $2$ | $1$ |
| 48.288.8-48.ju.1.1 | $48$ | $3$ | $3$ | $8$ |
| 48.384.7-48.hz.1.7 | $48$ | $4$ | $4$ | $7$ |
| 240.192.1-240.bbt.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbx.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcj.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcn.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdj.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdr.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bep.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bex.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.480.16-240.ha.2.6 | $240$ | $5$ | $5$ | $16$ |