Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.24.0.133 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{7}\cdot\frac{(x+y)^{24}(2401x^{8}+41160x^{6}y^{2}+26264x^{4}y^{4}+3360x^{2}y^{6}+16y^{8})^{3}}{y^{2}x^{2}(x+y)^{24}(7x^{2}-2y^{2})^{8}(7x^{2}+2y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|
| $X_0(8)$ | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0.s.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.12.0.bb.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 |
|---|---|---|---|---|
| 56.48.0.bi.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.bi.2 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.bj.1 | $56$ | $2$ | $2$ | $0$ |
| 56.48.0.bj.2 | $56$ | $2$ | $2$ | $0$ |
| 56.192.11.dx.1 | $56$ | $8$ | $8$ | $11$ |
| 56.504.34.fb.1 | $56$ | $21$ | $21$ | $34$ |
| 56.672.45.ff.1 | $56$ | $28$ | $28$ | $45$ |
| 112.48.0.w.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0.w.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0.x.1 | $112$ | $2$ | $2$ | $0$ |
| 112.48.0.x.2 | $112$ | $2$ | $2$ | $0$ |
| 112.48.1.r.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.t.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.cf.1 | $112$ | $2$ | $2$ | $1$ |
| 112.48.1.ch.1 | $112$ | $2$ | $2$ | $1$ |
| 168.48.0.dt.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.dt.2 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.du.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0.du.2 | $168$ | $2$ | $2$ | $0$ |
| 168.72.4.it.1 | $168$ | $3$ | $3$ | $4$ |
| 168.96.3.lr.1 | $168$ | $4$ | $4$ | $3$ |
| 280.48.0.dq.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.dq.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.dr.1 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0.dr.2 | $280$ | $2$ | $2$ | $0$ |
| 280.120.8.dt.1 | $280$ | $5$ | $5$ | $8$ |
| 280.144.7.he.1 | $280$ | $6$ | $6$ | $7$ |
| 280.240.15.jf.1 | $280$ | $10$ | $10$ | $15$ |