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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 56.96.0.1026 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}27&21\\16&29\end{bmatrix}$, $\begin{bmatrix}35&6\\12&9\end{bmatrix}$, $\begin{bmatrix}55&14\\20&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.bc.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| 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 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3\cdot5^2\cdot7^2}\cdot\frac{(7x-6y)^{48}(1034831x^{8}+2535456x^{7}y-35265888x^{6}y^{2}+126531328x^{5}y^{3}-198916480x^{4}y^{4}+95434752x^{3}y^{5}+66332672x^{2}y^{6}-134987776xy^{7}+76476416y^{8})^{3}(44829071x^{8}-90431264x^{7}y+50785952x^{6}y^{2}+83505408x^{5}y^{3}-198916480x^{4}y^{4}+144607232x^{3}y^{5}-46061568x^{2}y^{6}+3784704xy^{7}+1765376y^{8})^{3}}{(7x-6y)^{48}(7x^{2}-8y^{2})^{2}(7x^{2}-32xy+8y^{2})^{4}(7x^{2}-7xy+8y^{2})^{2}(2793x^{4}-3136x^{3}y+2352x^{2}y^{2}-3584xy^{3}+3648y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.k.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.k.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.bu.2.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.bu.2.15 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.bv.2.1 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.bv.2.12 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.768.23-56.dt.1.1 | $56$ | $8$ | $8$ | $23$ |
| 56.2016.70-56.fs.2.7 | $56$ | $21$ | $21$ | $70$ |
| 56.2688.93-56.fs.1.5 | $56$ | $28$ | $28$ | $93$ |
| 112.192.1-112.s.1.8 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.t.1.8 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.u.1.7 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.x.1.7 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.y.2.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bb.2.5 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bc.2.1 | $112$ | $2$ | $2$ | $1$ |
| 112.192.1-112.bd.2.1 | $112$ | $2$ | $2$ | $1$ |
| 168.288.8-168.pp.2.1 | $168$ | $3$ | $3$ | $8$ |
| 168.384.7-168.jy.2.9 | $168$ | $4$ | $4$ | $7$ |
| 280.480.16-280.ed.1.1 | $280$ | $5$ | $5$ | $16$ |