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 | $4^{6}$ | 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.303 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&32\\44&31\end{bmatrix}$, $\begin{bmatrix}13&28\\26&29\end{bmatrix}$, $\begin{bmatrix}13&48\\18&47\end{bmatrix}$, $\begin{bmatrix}23&52\\2&29\end{bmatrix}$, $\begin{bmatrix}43&12\\24&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.0.e.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| 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 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{3^4\cdot7^2}\cdot\frac{x^{24}(2401x^{8}+222264x^{4}y^{4}+104976y^{8})^{3}}{y^{4}x^{28}(7x^{2}-18y^{2})^{4}(7x^{2}+18y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-4.b.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 |
|---|---|---|---|---|
| 56.96.0-56.k.1.11 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.k.1.14 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.k.2.11 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.k.2.14 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.l.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.l.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.l.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.l.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.m.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.m.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.m.2.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.m.2.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.n.1.11 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.n.1.14 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.n.2.11 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.n.2.14 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-56.r.1.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.r.1.16 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.ba.1.4 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.ba.1.14 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bt.1.6 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bt.1.12 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bv.1.8 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bv.1.10 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.h.1.1 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.h.1.5 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.h.1.32 | $56$ | $28$ | $28$ | $45$ |
| 168.96.0-168.t.1.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.t.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.t.2.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.t.2.22 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.u.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.u.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.u.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.u.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.v.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.v.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.v.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.v.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.w.1.17 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.w.1.28 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.w.2.17 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.w.2.28 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-168.bv.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bv.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.by.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.by.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fb.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.fb.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ff.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ff.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.144.4-168.e.1.56 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.cu.1.57 | $168$ | $4$ | $4$ | $3$ |
| 280.96.0-280.u.1.18 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.u.1.21 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.u.2.18 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.u.2.19 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.v.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.v.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.v.2.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.v.2.14 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.w.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.w.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.w.2.1 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.w.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.x.1.19 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.x.1.21 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.x.2.18 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.x.2.21 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.bv.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.bv.1.12 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.by.1.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.by.1.12 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ex.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ex.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fb.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fb.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-280.e.1.25 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-280.g.1.49 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-280.e.1.54 | $280$ | $10$ | $10$ | $15$ |