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: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.0.804 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&4\\50&17\end{bmatrix}$, $\begin{bmatrix}15&36\\54&35\end{bmatrix}$, $\begin{bmatrix}39&32\\0&9\end{bmatrix}$, $\begin{bmatrix}47&32\\10&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.n.2 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 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^4\cdot7^2}\cdot\frac{x^{48}(5764801x^{16}+474360768x^{14}y^{2}+153692888832x^{12}y^{4}+5972066537472x^{10}y^{6}+107755294187520x^{8}y^{8}+631820263882752x^{6}y^{10}+1720250130235392x^{4}y^{12}+561714328240128x^{2}y^{14}+722204136308736y^{16})^{3}}{y^{4}x^{52}(7x^{2}-72y^{2})^{8}(7x^{2}+72y^{2})^{4}(49x^{4}+3024x^{2}y^{2}+5184y^{4})^{4}}$ |
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.1.10 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.e.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.e.1.17 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.18 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.