Invariants
| Level: | $57$ | $\SL_2$-level: | $3$ | ||||
| Index: | $8$ | $\PSL_2$-index: | $4$ | ||||
| Genus: | $0 = 1 + \frac{ 4 }{12} - \frac{ 0 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot3$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 3B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 57.8.0.1 |
Level structure
| $\GL_2(\Z/57\Z)$-generators: | $\begin{bmatrix}9&43\\8&4\end{bmatrix}$, $\begin{bmatrix}56&39\\38&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 3.4.0.a.1 for the level structure with $-I$) |
| Cyclic 57-isogeny field degree: | $20$ |
| Cyclic 57-torsion field degree: | $720$ |
| Full 57-torsion field degree: | $738720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 78278 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 4 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{x^{4}(x-18y)^{3}(x+30y)}{y^{3}x^{4}(x-24y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 57.24.0-3.a.1.1 | $57$ | $3$ | $3$ | $0$ |
| 114.16.0-6.a.1.2 | $114$ | $2$ | $2$ | $0$ |
| 114.16.0-6.b.1.2 | $114$ | $2$ | $2$ | $0$ |
| 114.24.0-6.a.1.4 | $114$ | $3$ | $3$ | $0$ |
| 171.24.0-9.a.1.2 | $171$ | $3$ | $3$ | $0$ |
| 171.24.0-9.b.1.2 | $171$ | $3$ | $3$ | $0$ |
| 171.24.1-9.a.1.2 | $171$ | $3$ | $3$ | $1$ |
| 228.16.0-12.a.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.16.0-12.b.1.2 | $228$ | $2$ | $2$ | $0$ |
| 228.32.1-12.a.1.8 | $228$ | $4$ | $4$ | $1$ |
| 285.40.1-15.a.1.4 | $285$ | $5$ | $5$ | $1$ |
| 285.48.1-15.a.1.1 | $285$ | $6$ | $6$ | $1$ |
| 285.80.2-15.a.1.8 | $285$ | $10$ | $10$ | $2$ |
| 57.160.5-57.a.1.8 | $57$ | $20$ | $20$ | $5$ |
| 57.1368.49-57.a.1.7 | $57$ | $171$ | $171$ | $49$ |
| 57.1520.54-57.a.1.4 | $57$ | $190$ | $190$ | $54$ |
| 57.2280.79-57.a.1.4 | $57$ | $285$ | $285$ | $79$ |
| 114.16.0-114.a.1.2 | $114$ | $2$ | $2$ | $0$ |
| 114.16.0-114.b.1.2 | $114$ | $2$ | $2$ | $0$ |
| 228.16.0-228.a.1.7 | $228$ | $2$ | $2$ | $0$ |
| 228.16.0-228.b.1.6 | $228$ | $2$ | $2$ | $0$ |