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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.338 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot7^4}\cdot\frac{(2x+y)^{48}(23750311936x^{16}+439667392512x^{15}y+3613787029504x^{14}y^{2}+17439258181632x^{13}y^{3}+55908421976064x^{12}y^{4}+127040888733696x^{11}y^{5}+211887120392192x^{10}y^{6}+264093484425216x^{9}y^{7}+247602642257408x^{8}y^{8}+173888969926656x^{7}y^{9}+89664455111168x^{6}y^{10}+32252814429696x^{5}y^{11}+7115392127424x^{4}y^{12}+619826481792x^{3}y^{13}-19552871456x^{2}y^{14}+32189964000xy^{15}+11296844833y^{16})^{3}}{(2x+y)^{48}(2x^{2}+3xy+2y^{2})^{4}(4x^{2}-4xy-7y^{2})^{8}(4x^{2}+20xy-3y^{2})^{2}(20x^{2}+44xy+13y^{2})^{8}(36x^{2}+68xy+29y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0.e.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0.h.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0.m.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.h.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.i.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.x.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.y.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bk.2 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bl.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bo.1 | $56$ | $2$ | $2$ | $1$ |
| 56.96.1.bp.1 | $56$ | $2$ | $2$ | $1$ |
| 56.384.23.cw.1 | $56$ | $8$ | $8$ | $23$ |
| 56.1008.70.ep.2 | $56$ | $21$ | $21$ | $70$ |
| 56.1344.93.ep.2 | $56$ | $28$ | $28$ | $93$ |
| 168.96.1.iy.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.iz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.je.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.jf.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ke.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.kf.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.kk.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.kl.1 | $168$ | $2$ | $2$ | $1$ |
| 168.144.8.nt.2 | $168$ | $3$ | $3$ | $8$ |
| 168.192.7.ht.2 | $168$ | $4$ | $4$ | $7$ |
| 280.96.1.iy.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.iz.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.je.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.jf.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.ke.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.kf.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.kk.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.kl.2 | $280$ | $2$ | $2$ | $1$ |
| 280.240.16.df.2 | $280$ | $5$ | $5$ | $16$ |
| 280.288.15.ia.1 | $280$ | $6$ | $6$ | $15$ |