Invariants
| Level: | $256$ | $\SL_2$-level: | $256$ | Newform level: | $1$ | ||
| Index: | $1572864$ | $\PSL_2$-index: | $1572864$ | ||||
| Genus: | $125953 = 1 + \frac{ 1572864 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10240 }{2}$ | ||||||
| Cusps: | $10240$ (none of which are rational) | Cusp widths | $64^{4096}\cdot128^{2048}\cdot256^{4096}$ | Cusp orbits | $32^{4}\cdot64^{96}\cdot128^{31}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $15600 \le \gamma \le 251904$ | ||||||
| $\overline{\Q}$-gonality: | $15600 \le \gamma \le 125953$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Level structure
| $\GL_2(\Z/256\Z)$-generators: | $\begin{bmatrix}57&26\\0&251\end{bmatrix}$, $\begin{bmatrix}121&124\\0&165\end{bmatrix}$, $\begin{bmatrix}143&98\\0&121\end{bmatrix}$, $\begin{bmatrix}175&114\\0&169\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 256.3145728.125953-256.j.1.1, 256.3145728.125953-256.j.1.2, 256.3145728.125953-256.j.1.3, 256.3145728.125953-256.j.1.4, 256.3145728.125953-256.j.1.5, 256.3145728.125953-256.j.1.6 |
| Cyclic 256-isogeny field degree: | $1$ |
| Cyclic 256-torsion field degree: | $64$ |
| Full 256-torsion field degree: | $1024$ |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,7,\ldots,2297$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 256.786432.62465.i.5 | $256$ | $2$ | $2$ | $62465$ | $?$ |
| 256.786432.62465.j.5 | $256$ | $2$ | $2$ | $62465$ | $?$ |
| 256.786432.62465.bv.8 | $256$ | $2$ | $2$ | $62465$ | $?$ |
| 256.786432.62465.bx.1 | $256$ | $2$ | $2$ | $62465$ | $?$ |
| 256.786432.62465.by.3 | $256$ | $2$ | $2$ | $62465$ | $?$ |
| 256.786432.62977.j.7 | $256$ | $2$ | $2$ | $62977$ | $?$ |
| 256.786432.62977.bp.2 | $256$ | $2$ | $2$ | $62977$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 256.3145728.253953.e.1 | $256$ | $2$ | $2$ | $253953$ |
| 256.3145728.253953.e.3 | $256$ | $2$ | $2$ | $253953$ |