Invariants
| Level: | $256$ | $\SL_2$-level: | $256$ | Newform level: | $1$ | ||
| Index: | $3145728$ | $\PSL_2$-index: | $3145728$ | ||||
| Genus: | $253953 = 1 + \frac{ 3145728 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16384 }{2}$ | ||||||
| Cusps: | $16384$ (none of which are rational) | Cusp widths | $128^{8192}\cdot256^{8192}$ | Cusp orbits | $32^{4}\cdot64^{128}\cdot128^{63}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $31200 \le \gamma \le 507904$ | ||||||
| $\overline{\Q}$-gonality: | $31200 \le \gamma \le 253953$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Level structure
| $\GL_2(\Z/256\Z)$-generators: | $\begin{bmatrix}1&128\\0&1\end{bmatrix}$, $\begin{bmatrix}247&126\\0&205\end{bmatrix}$, $\begin{bmatrix}255&128\\0&255\end{bmatrix}$, $\begin{bmatrix}255&250\\0&1\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 256.6291456.253953-256.e.1.1, 256.6291456.253953-256.e.1.2, 256.6291456.253953-256.e.1.3, 256.6291456.253953-256.e.1.4, 256.6291456.253953-256.e.1.5, 256.6291456.253953-256.e.1.6 |
| Cyclic 256-isogeny field degree: | $1$ |
| Cyclic 256-torsion field degree: | $64$ |
| Full 256-torsion field degree: | $512$ |
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 |
|---|---|---|---|---|---|
| 256.1572864.125953.i.1 | $256$ | $2$ | $2$ | $125953$ | $?$ |
| 256.1572864.125953.j.1 | $256$ | $2$ | $2$ | $125953$ | $?$ |
| 256.1572864.125953.bo.8 | $256$ | $2$ | $2$ | $125953$ | $?$ |
| 256.1572864.125953.bp.3 | $256$ | $2$ | $2$ | $125953$ | $?$ |
| 256.1572864.126977.e.7 | $256$ | $2$ | $2$ | $126977$ | $?$ |
| 256.1572864.126977.bc.6 | $256$ | $2$ | $2$ | $126977$ | $?$ |
| 256.1572864.126977.be.1 | $256$ | $2$ | $2$ | $126977$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 256.6291456.512001.e.2 | $256$ | $2$ | $2$ | $512001$ |
| 256.6291456.512001.e.3 | $256$ | $2$ | $2$ | $512001$ |