Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 8C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.12.0.7 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 596 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^8\cdot7}\cdot\frac{x^{12}(49x^{4}+8064x^{2}y^{2}+82944y^{4})^{3}}{y^{8}x^{14}(7x^{2}+1152y^{2})}$ |
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 |
|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.24.0.l.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.o.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.bb.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.bc.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.be.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.bh.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.br.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0.bs.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.5.bp.1 | $56$ | $8$ | $8$ | $5$ |
| 56.252.16.cv.1 | $56$ | $21$ | $21$ | $16$ |
| 56.336.21.cv.1 | $56$ | $28$ | $28$ | $21$ |
| 168.24.0.bv.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.bx.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.cd.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.cf.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.di.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.dl.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.dv.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0.dw.1 | $168$ | $2$ | $2$ | $0$ |
| 168.36.2.dj.1 | $168$ | $3$ | $3$ | $2$ |
| 168.48.1.zz.1 | $168$ | $4$ | $4$ | $1$ |
| 280.24.0.cb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.cd.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.cj.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.cl.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.do.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.dr.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.eb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ec.1 | $280$ | $2$ | $2$ | $0$ |
| 280.60.4.cd.1 | $280$ | $5$ | $5$ | $4$ |
| 280.72.3.ct.1 | $280$ | $6$ | $6$ | $3$ |
| 280.120.7.dj.1 | $280$ | $10$ | $10$ | $7$ |