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: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.124 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&34\\28&55\end{bmatrix}$, $\begin{bmatrix}16&43\\13&34\end{bmatrix}$, $\begin{bmatrix}22&35\\15&26\end{bmatrix}$, $\begin{bmatrix}28&51\\19&44\end{bmatrix}$, $\begin{bmatrix}40&7\\11&52\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| 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 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.12.0-4.c.1.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0-4.c.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.48.0-8.i.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.k.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.q.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.r.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.ba.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.ba.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.bb.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-8.bb.2.4 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bf.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bh.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bj.1.9 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bl.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bu.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bu.2.13 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bv.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0-56.bv.2.11 | $56$ | $2$ | $2$ | $0$ |
| 56.192.5-56.bl.1.42 | $56$ | $8$ | $8$ | $5$ |
| 56.504.16-56.cj.1.46 | $56$ | $21$ | $21$ | $16$ |
| 56.672.21-56.cj.1.22 | $56$ | $28$ | $28$ | $21$ |
| 112.48.0-16.e.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.e.2.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.e.1.18 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.e.2.26 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.f.1.3 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.f.2.7 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.f.1.18 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.f.2.22 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.g.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.g.1.24 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-16.h.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0-112.h.1.20 | $112$ | $2$ | $2$ | $0$ |
| 112.48.1-16.a.1.11 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.a.1.13 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-16.b.1.11 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1-112.b.1.9 | $112$ | $2$ | $2$ | $1$ |
| 168.48.0-24.bh.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bj.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bl.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bn.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.by.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.by.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bz.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-24.bz.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cx.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cz.1.23 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.db.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dd.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ec.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ec.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ed.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ed.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.72.2-24.cj.1.8 | $168$ | $3$ | $3$ | $2$ |
| 168.96.1-24.ir.1.6 | $168$ | $4$ | $4$ | $1$ |
| 280.48.0-40.bj.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bl.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bn.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.bp.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.ca.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.ca.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.cb.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-40.cb.2.12 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dd.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.df.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dh.1.20 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dj.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ei.1.18 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ei.2.27 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ej.1.19 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ej.2.22 | $280$ | $2$ | $2$ | $0$ |
| 280.120.4-40.bl.1.13 | $280$ | $5$ | $5$ | $4$ |
| 280.144.3-40.bx.1.43 | $280$ | $6$ | $6$ | $3$ |
| 280.240.7-40.cj.1.46 | $280$ | $10$ | $10$ | $7$ |