Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.724 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&30\\30&31\end{bmatrix}$, $\begin{bmatrix}24&15\\33&2\end{bmatrix}$, $\begin{bmatrix}24&37\\29&0\end{bmatrix}$, $\begin{bmatrix}42&13\\1&46\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.ba.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $64512$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 138 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{x^{24}(x^{8}+4x^{6}y^{2}-10x^{4}y^{4}-28x^{2}y^{6}+y^{8})^{3}}{y^{4}x^{26}(x^{2}+y^{2})^{8}(x^{2}+2y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.24.0-8.n.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.n.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.96.0-8.j.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.m.2.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.n.2.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.o.1.2 | $56$ | $2$ | $2$ | $0$ |
| 112.96.0-16.u.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.w.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.y.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.ba.1.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-16.q.1.4 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.s.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.u.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.w.1.4 | $112$ | $2$ | $2$ | $1$ |
| 168.96.0-24.bi.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bk.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bm.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bo.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.144.4-24.ge.1.29 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-24.gf.2.24 | $168$ | $4$ | $4$ | $3$ |
| 280.96.0-40.bi.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.bk.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.bm.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.bo.2.1 | $280$ | $2$ | $2$ | $0$ |
| 280.240.8-40.da.2.5 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-40.fn.2.5 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-40.gq.2.17 | $280$ | $10$ | $10$ | $15$ |
| 56.96.0-56.bg.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bi.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bk.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bm.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.384.11-56.fb.2.12 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.ge.1.4 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.gq.2.9 | $56$ | $28$ | $28$ | $45$ |
| 112.96.0-112.be.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bg.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bm.2.1 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.bo.2.5 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-112.bq.2.12 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.bs.2.16 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.by.2.12 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.ca.1.10 | $112$ | $2$ | $2$ | $1$ |
| 168.96.0-168.dz.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ed.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.eh.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.el.1.5 | $168$ | $2$ | $2$ | $0$ |
| 280.96.0-280.dw.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ea.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ee.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ei.1.5 | $280$ | $2$ | $2$ | $0$ |