Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.320 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^4\cdot7^2}\cdot\frac{x^{48}(5764801x^{16}-237180384x^{14}y^{2}+38423222208x^{12}y^{4}-746508317184x^{10}y^{6}+6734705886720x^{8}y^{8}-19744383246336x^{6}y^{10}+26878908284928x^{4}y^{12}-4388393189376x^{2}y^{14}+2821109907456y^{16})^{3}}{y^{4}x^{52}(7x^{2}-36y^{2})^{4}(7x^{2}+36y^{2})^{8}(49x^{4}-1512x^{2}y^{2}+1296y^{4})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0.e.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 28.24.0.c.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0.h.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 |
|---|---|---|---|---|
| 56.96.1.i.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.y.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bc.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bg.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bu.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.by.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.cb.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.cd.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.23.x.1 | $56$ | $8$ | $8$ | $23$ |
| 56.1008.70.bd.2 | $56$ | $21$ | $21$ | $70$ |
| 56.1344.93.bd.1 | $56$ | $28$ | $28$ | $93$ |
| 168.96.1.fx.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.gd.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.hc.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.hi.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.me.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.mk.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.nk.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.nq.2 | $168$ | $2$ | $2$ | $1$ |
| 168.144.8.bz.2 | $168$ | $3$ | $3$ | $8$ |
| 168.192.7.cf.2 | $168$ | $4$ | $4$ | $7$ |
| 280.96.1.fx.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.gd.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.hc.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.hi.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.lk.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.lq.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.mq.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.mw.1 | $280$ | $2$ | $2$ | $1$ |
| 280.240.16.bb.2 | $280$ | $5$ | $5$ | $16$ |
| 280.288.15.ci.2 | $280$ | $6$ | $6$ | $15$ |