Invariants
| Level: | $32$ | $\SL_2$-level: | $32$ | Newform level: | $1024$ | ||
| Index: | $512$ | $\PSL_2$-index: | $512$ | ||||
| Genus: | $35 = 1 + \frac{ 512 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $32^{16}$ | Cusp orbits | $16$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | $14$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 32.512.35.1 |
Level structure
| $\GL_2(\Z/32\Z)$-generators: | $\begin{bmatrix}9&4\\28&5\end{bmatrix}$, $\begin{bmatrix}15&5\\27&10\end{bmatrix}$, $\begin{bmatrix}31&0\\0&31\end{bmatrix}$ |
| $\GL_2(\Z/32\Z)$-subgroup: | $C_2\times C_8\times C_{48}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 32.1024.35-32.a.1.1, 32.1024.35-32.a.1.2 |
| Cyclic 32-isogeny field degree: | $48$ |
| Cyclic 32-torsion field degree: | $768$ |
| Full 32-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{334}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{5}\cdot2^{7}\cdot4^{4}$ |
| Newforms: | 64.2.a.a, 256.2.a.a, 256.2.a.b, 256.2.a.c, 256.2.a.d, 256.2.a.e, 1024.2.a.a, 1024.2.a.b, 1024.2.a.c, 1024.2.a.d, 1024.2.a.e, 1024.2.a.f, 1024.2.a.g, 1024.2.a.h, 1024.2.a.i, 1024.2.a.j |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(16)$ | $16$ | $4$ | $4$ | $7$ | $2$ | $2^{6}\cdot4^{4}$ |
| $X_{\mathrm{ns}}^+(32)$ | $32$ | $2$ | $2$ | $14$ | $14$ | $1^{3}\cdot2^{5}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 32.1536.105.a.2 | $32$ | $3$ | $3$ | $105$ | $42$ | $1^{10}\cdot2^{14}\cdot4^{8}$ |
| $X_{\mathrm{ns}}(64)$ | $64$ | $4$ | $4$ | $155$ | $70$ | $4^{8}\cdot8^{11}$ |