Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.65 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}26&43\\7&22\end{bmatrix}$, $\begin{bmatrix}27&36\\48&35\end{bmatrix}$, $\begin{bmatrix}28&29\\9&36\end{bmatrix}$, $\begin{bmatrix}49&6\\38&21\end{bmatrix}$, $\begin{bmatrix}50&43\\39&26\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.d.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $129024$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 746 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{(4x-y)^{12}(64x^{2}-32xy+y^{2})^{3}(64x^{2}+32xy+y^{2})^{3}}{y^{2}x^{2}(4x-y)^{12}(64x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.48.0-4.c.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-28.d.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.p.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.q.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.t.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.t.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.u.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.w.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.x.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.x.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.ba.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bb.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.48.1-8.m.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-56.m.1.1 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-8.n.1.3 | $56$ | $2$ | $2$ | $1$ |
| 56.48.1-56.n.1.5 | $56$ | $2$ | $2$ | $1$ |
| 56.192.5-28.h.1.29 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-28.p.1.13 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-28.p.1.22 | $56$ | $28$ | $28$ | $21$ |
| 168.48.0-12.e.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.h.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.u.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.v.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.z.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bc.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bd.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bf.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bg.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bj.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bm.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.bn.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-24.m.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.m.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-24.n.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.n.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.72.2-12.p.1.13 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-12.h.1.18 | $168$ | $4$ | $4$ | $1$ |
| 280.48.0-20.d.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.h.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.v.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.w.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.z.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.be.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bf.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bh.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bi.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bl.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bq.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.br.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-40.m.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.m.1.9 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-40.n.1.7 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.n.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.120.4-20.h.1.9 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-20.l.1.17 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-20.p.1.11 | $280$ | $10$ | $10$ | $7$ |