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 | $4^{8}\cdot8^{2}$ | 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: | 56.96.0.802 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&28\\34&33\end{bmatrix}$, $\begin{bmatrix}5&4\\46&19\end{bmatrix}$, $\begin{bmatrix}19&36\\44&45\end{bmatrix}$, $\begin{bmatrix}35&4\\46&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.0.j.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^8\cdot7^2}\cdot\frac{(x+y)^{48}(10484158039713x^{16}-122695552305648x^{15}y+955341733507056x^{14}y^{2}-8632224393145920x^{13}y^{3}+78752156398509888x^{12}y^{4}-560698436644488576x^{11}y^{5}+2889974568982906944x^{10}y^{6}-10721243747284076160x^{9}y^{7}+29336598453951502560x^{8}y^{8}-64923142420402801920x^{7}y^{9}+144008658212010552576x^{6}y^{10}-374390523149276648448x^{5}y^{11}+979998687127352890368x^{4}y^{12}-2085115427909179545600x^{3}y^{13}+3163171558478413999104x^{2}y^{14}-3000539209708342278144xy^{15}+1335450362348513329408y^{16})^{3}}{(x+y)^{48}(3x-8y)^{4}(3x+14y)^{4}(9x^{2}-70xy+42y^{2})^{8}(18x^{2}-63xy+161y^{2})^{4}(1863x^{4}-6804x^{3}y+27216x^{2}y^{2}-109368xy^{3}+191492y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-28.c.1.12 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-28.c.1.16 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-8.e.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.2.14 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.48.0-56.i.2.20 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.