Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| 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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.0.805 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}19&12\\50&41\end{bmatrix}$, $\begin{bmatrix}23&44\\44&47\end{bmatrix}$, $\begin{bmatrix}31&32\\8&45\end{bmatrix}$, $\begin{bmatrix}33&20\\8&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.v.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $32256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2\cdot3^8\cdot7^4}\cdot\frac{(3x+y)^{48}(14085801767927808x^{16}+89733863936360448x^{15}y+277921329076371456x^{14}y^{2}+551708791768350720x^{13}y^{3}+779105635806855168x^{12}y^{4}+822996333140115456x^{11}y^{5}+667927763264692224x^{10}y^{6}+422622716216279040x^{9}y^{7}+210036643626862080x^{8}y^{8}+82176639264276480x^{7}y^{9}+25253441666643456x^{6}y^{10}+6050405998522368x^{5}y^{11}+1113726370534848x^{4}y^{12}+153351349079040x^{3}y^{13}+15020870899104x^{2}y^{14}+943029206784xy^{15}+28783651393y^{16})^{3}}{(3x+y)^{48}(18x^{2}+21xy+7y^{2})^{2}(36x^{2}-7y^{2})^{4}(36x^{2}+28xy+7y^{2})^{8}(36x^{2}+36xy+7y^{2})^{8}(72x^{2}+42xy+7y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.e.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.2.20 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.2.23 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.m.1.14 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.m.1.19 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.