Invariants
| Level: | $56$ | $\SL_2$-level: | $14$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot14$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 14B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.32.0.5 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}23&27\\0&53\end{bmatrix}$, $\begin{bmatrix}34&51\\45&35\end{bmatrix}$, $\begin{bmatrix}39&55\\31&4\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.16.0.d.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $12$ |
| Cyclic 56-torsion field degree: | $288$ |
| Full 56-torsion field degree: | $96768$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^{21}\cdot3^{14}}\cdot\frac{x^{16}(7x^{4}-936x^{2}y^{2}+36288y^{4})(49x^{4}-2520x^{2}y^{2}+5184y^{4})^{3}}{y^{14}x^{18}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 7.16.0-7.a.1.1 | $7$ | $2$ | $2$ | $0$ | $0$ |
| 56.16.0-7.a.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.96.2-56.k.1.3 | $56$ | $3$ | $3$ | $2$ |
| 56.96.2-56.k.2.7 | $56$ | $3$ | $3$ | $2$ |
| 56.96.2-56.m.1.3 | $56$ | $3$ | $3$ | $2$ |
| 56.96.2-56.q.1.2 | $56$ | $3$ | $3$ | $2$ |
| 56.128.3-56.d.1.7 | $56$ | $4$ | $4$ | $3$ |
| 56.224.5-56.bf.1.4 | $56$ | $7$ | $7$ | $5$ |
| 168.96.4-168.f.1.1 | $168$ | $3$ | $3$ | $4$ |
| 168.128.3-168.h.1.2 | $168$ | $4$ | $4$ | $3$ |
| 280.160.4-280.f.1.13 | $280$ | $5$ | $5$ | $4$ |
| 280.192.7-280.j.1.5 | $280$ | $6$ | $6$ | $7$ |
| 280.320.11-280.f.1.17 | $280$ | $10$ | $10$ | $11$ |