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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.256 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&32\\12&23\end{bmatrix}$, $\begin{bmatrix}23&48\\24&47\end{bmatrix}$, $\begin{bmatrix}33&4\\54&41\end{bmatrix}$, $\begin{bmatrix}45&32\\36&33\end{bmatrix}$, $\begin{bmatrix}55&32\\6&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.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: | $64512$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 136 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{x^{24}(256x^{8}+256x^{6}y^{2}+80x^{4}y^{4}+8x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(2x^{2}+y^{2})^{2}(4x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.24.0-4.b.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 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 |
|---|---|---|---|---|
| 56.96.0-8.a.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.b.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.d.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.e.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.g.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.g.1.7 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.h.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.h.2.9 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.j.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.k.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.k.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.l.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.o.1.13 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.p.2.13 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.s.2.16 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.t.1.16 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-8.e.2.6 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-8.i.1.4 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-8.l.1.8 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-8.m.2.8 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bc.2.13 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bd.2.5 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bg.2.7 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bh.1.8 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.p.2.50 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.s.2.60 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.u.1.56 | $56$ | $28$ | $28$ | $45$ |
| 168.96.0-24.g.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.h.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.k.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.l.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.p.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.q.2.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.t.2.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.u.2.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.x.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.z.2.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bf.2.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bh.2.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bn.2.28 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bp.1.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bv.2.17 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bx.1.30 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-24.bc.2.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bd.2.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bg.2.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bh.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ds.2.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.du.2.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ea.2.22 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ec.2.12 | $168$ | $2$ | $2$ | $1$ |
| 168.144.4-24.s.2.61 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-24.bn.2.64 | $168$ | $4$ | $4$ | $3$ |
| 280.96.0-40.i.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.j.2.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.m.2.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.n.2.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.q.2.14 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.r.2.16 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.u.2.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.v.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.y.2.10 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.ba.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bg.2.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bi.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bo.1.24 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bq.2.31 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bw.1.22 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.by.2.23 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-40.bc.2.10 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.bd.2.16 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.bg.2.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.bh.2.10 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.do.2.23 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dq.2.7 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dw.2.13 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dy.2.12 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-40.k.2.31 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-40.q.2.56 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-40.s.2.63 | $280$ | $10$ | $10$ | $15$ |