Invariants
| Level: | $64$ | $\SL_2$-level: | $64$ | Newform level: | $4096$ | ||
| Index: | $6144$ | $\PSL_2$-index: | $3072$ | ||||
| Genus: | $225 = 1 + \frac{ 3072 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 64 }{2}$ | ||||||
| Cusps: | $64$ (none of which are rational) | Cusp widths | $32^{32}\cdot64^{32}$ | Cusp orbits | $16^{2}\cdot32$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $101$ | ||||||
| $\Q$-gonality: | $52 \le \gamma \le 64$ | ||||||
| $\overline{\Q}$-gonality: | $52 \le \gamma \le 64$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 64.6144.225.4133 |
Level structure
| $\GL_2(\Z/64\Z)$-generators: | $\begin{bmatrix}3&26\\36&47\end{bmatrix}$, $\begin{bmatrix}45&15\\32&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 64.3072.225.kf.1 for the level structure with $-I$) |
| Cyclic 64-isogeny field degree: | $16$ |
| Cyclic 64-torsion field degree: | $256$ |
| Full 64-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{2497}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{8}\cdot4^{14}\cdot8^{18}$ |
| Newforms: | 32.2.a.a, 128.2.a.b$^{2}$, 128.2.a.d$^{2}$, 256.2.a.a$^{3}$, 256.2.a.d, 512.2.a.b$^{2}$, 512.2.a.e$^{2}$, 512.2.a.g$^{2}$, 1024.2.a.b$^{3}$, 1024.2.a.e, 1024.2.a.g$^{3}$, 1024.2.a.j, 2048.2.a.a$^{2}$, 2048.2.a.b$^{2}$, 2048.2.a.d$^{2}$, 2048.2.a.f$^{2}$, 2048.2.a.h$^{2}$, 4096.2.a.e$^{3}$, 4096.2.a.f, 4096.2.a.i, 4096.2.a.k$^{3}$, 4096.2.a.n, 4096.2.a.o$^{3}$, 4096.2.a.q, 4096.2.a.s$^{3}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,11,\ldots,409$, 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 |
|---|---|---|---|---|---|---|
| 64.3072.105-64.li.1.1 | $64$ | $2$ | $2$ | $105$ | $49$ | $4^{6}\cdot8^{12}$ |
| 64.3072.105-64.li.1.2 | $64$ | $2$ | $2$ | $105$ | $49$ | $4^{6}\cdot8^{12}$ |
| 64.3072.105-64.th.1.1 | $64$ | $2$ | $2$ | $105$ | $53$ | $4^{6}\cdot8^{12}$ |
| 64.3072.105-64.th.1.3 | $64$ | $2$ | $2$ | $105$ | $53$ | $4^{6}\cdot8^{12}$ |
| 64.3072.113-64.jf.1.1 | $64$ | $2$ | $2$ | $113$ | $49$ | $4^{4}\cdot8^{12}$ |
| 64.3072.113-64.jf.1.3 | $64$ | $2$ | $2$ | $113$ | $49$ | $4^{4}\cdot8^{12}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 128.12288.465-128.ma.1.2 | $128$ | $2$ | $2$ | $465$ | $?$ | not computed |
| 128.12288.465-128.ma.1.3 | $128$ | $2$ | $2$ | $465$ | $?$ | not computed |
| 128.12288.465-128.me.1.1 | $128$ | $2$ | $2$ | $465$ | $?$ | not computed |
| 128.12288.465-128.mi.1.2 | $128$ | $2$ | $2$ | $465$ | $?$ | not computed |
| 128.12288.465-128.mi.1.3 | $128$ | $2$ | $2$ | $465$ | $?$ | not computed |
| 128.12288.481-128.gc.1.2 | $128$ | $2$ | $2$ | $481$ | $?$ | not computed |