Invariants
| Level: | $56$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.12.0.65 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}0&19\\17&34\end{bmatrix}$, $\begin{bmatrix}34&5\\17&16\end{bmatrix}$, $\begin{bmatrix}42&37\\19&42\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $258048$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 18 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{7^2}\cdot\frac{(4x+y)^{12}(118784x^{4}+129024x^{3}y+60032x^{2}y^{2}+14112xy^{3}+1421y^{4})^{3}}{(4x+y)^{12}(64x^{2}+48xy+7y^{2})^{4}(192x^{2}+112xy+21y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.6.0.b.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 28.6.0.d.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 56.6.0.c.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.96.5.r.1 | $56$ | $8$ | $8$ | $5$ |
| 56.252.16.bd.1 | $56$ | $21$ | $21$ | $16$ |
| 56.336.21.bd.1 | $56$ | $28$ | $28$ | $21$ |
| 168.36.2.ba.1 | $168$ | $3$ | $3$ | $2$ |
| 168.48.1.jy.1 | $168$ | $4$ | $4$ | $1$ |
| 280.60.4.o.1 | $280$ | $5$ | $5$ | $4$ |
| 280.72.3.u.1 | $280$ | $6$ | $6$ | $3$ |
| 280.120.7.ba.1 | $280$ | $10$ | $10$ | $7$ |