Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.0.21 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&16\\50&25\end{bmatrix}$, $\begin{bmatrix}19&28\\10&13\end{bmatrix}$, $\begin{bmatrix}23&32\\46&3\end{bmatrix}$, $\begin{bmatrix}39&36\\46&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.v.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $32256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot3^4\cdot7^4}\cdot\frac{(x+2y)^{48}(2384487016353x^{16}-148195052330976x^{15}y+4744729430691264x^{14}y^{2}-96355486420784640x^{13}y^{3}+1411423453555008768x^{12}y^{4}-16407379517059786752x^{11}y^{5}+157522415555090589696x^{10}y^{6}-1248132319366184017920x^{9}y^{7}+8193743274148072058880x^{8}y^{8}-45240794048400948264960x^{7}y^{9}+208848769371005803167744x^{6}y^{10}-770893593178994553913344x^{5}y^{11}+2150227143065757929177088x^{4}y^{12}-4412525971045839053783040x^{3}y^{13}+6894546444959611770372096x^{2}y^{14}-8362262532516624048586752xy^{15}+6037856027692309915107328y^{16})^{3}}{(x+2y)^{48}(3x-16y)^{8}(3x+28y)^{8}(9x^{2}-140xy+168y^{2})^{4}(9x^{2}-63xy+322y^{2})^{8}(1863x^{4}-13608x^{3}y+108864x^{2}y^{2}-874944xy^{3}+3063872y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.e.2.11 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.19 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.1.24 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.m.1.13 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.m.1.17 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.