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 $4$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{4}\cdot2$ | ||
| 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: | 8J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.329 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&40\\18&47\end{bmatrix}$, $\begin{bmatrix}23&20\\42&11\end{bmatrix}$, $\begin{bmatrix}27&40\\0&13\end{bmatrix}$, $\begin{bmatrix}29&16\\24&31\end{bmatrix}$, $\begin{bmatrix}43&40\\24&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.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 220 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}-16x^{6}y^{2}+320x^{4}y^{4}-2048x^{2}y^{6}+4096y^{8})^{3}}{y^{4}x^{32}(x-2y)^{2}(x+2y)^{2}(x^{2}-8y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.24.0-4.b.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 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.b.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.c.1.9 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.e.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.f.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.h.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.i.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.i.2.7 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.j.2.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.k.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.l.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.m.2.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.n.2.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.q.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.r.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.u.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.v.1.8 | $56$ | $2$ | $2$ | $0$ |
| 56.96.1-8.i.2.6 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-8.k.2.4 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-8.m.2.6 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-8.n.1.7 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.be.2.10 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bf.2.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bi.2.10 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1-56.bj.2.12 | $56$ | $2$ | $2$ | $1$ |
| 56.384.11-56.s.2.14 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.z.2.7 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.bb.1.9 | $56$ | $28$ | $28$ | $45$ |
| 168.96.0-24.i.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.j.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.m.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.n.2.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.r.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.s.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.v.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.w.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bc.2.21 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.be.2.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bk.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bm.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bs.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.bu.1.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ca.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cc.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-24.be.2.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bf.2.10 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bi.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.bj.2.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dx.2.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.dz.2.23 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.ef.2.11 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.eh.2.2 | $168$ | $2$ | $2$ | $1$ |
| 168.144.4-24.z.2.2 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-24.bq.2.6 | $168$ | $4$ | $4$ | $3$ |
| 280.96.0-40.k.2.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.l.2.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.o.2.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.p.2.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.s.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.t.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.w.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.x.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bd.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bf.2.14 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bl.2.8 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bn.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bt.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.bv.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cb.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cd.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-40.be.2.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.bf.2.15 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.bi.2.11 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-40.bj.2.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dt.2.21 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.dv.2.5 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.eb.2.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.ed.2.11 | $280$ | $2$ | $2$ | $1$ |
| 280.240.8-40.n.2.1 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-40.v.2.4 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-40.z.2.4 | $280$ | $10$ | $10$ | $15$ |