Invariants
| Level: | $68$ | $\SL_2$-level: | $68$ | Newform level: | $1156$ | ||
| Index: | $1836$ | $\PSL_2$-index: | $918$ | ||||
| Genus: | $64 = 1 + \frac{ 918 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 27 }{2}$ | ||||||
| Cusps: | $27$ (of which $3$ are rational) | Cusp widths | $34^{27}$ | Cusp orbits | $1^{3}\cdot8^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $28$ | ||||||
| $\Q$-gonality: | $17 \le \gamma \le 42$ | ||||||
| $\overline{\Q}$-gonality: | $17 \le \gamma \le 42$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.1836.64.10 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}3&8\\20&65\end{bmatrix}$, $\begin{bmatrix}5&60\\40&63\end{bmatrix}$, $\begin{bmatrix}13&64\\54&17\end{bmatrix}$, $\begin{bmatrix}43&30\\54&47\end{bmatrix}$, $\begin{bmatrix}45&66\\64&57\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 34.918.64.a.1 for the level structure with $-I$) |
| Cyclic 68-isogeny field degree: | $4$ |
| Cyclic 68-torsion field degree: | $128$ |
| Full 68-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{58}\cdot17^{121}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{13}\cdot3^{8}\cdot4$ |
| Newforms: | 17.2.a.a$^{3}$, 34.2.a.a$^{2}$, 68.2.a.a, 289.2.a.a$^{3}$, 289.2.a.b$^{3}$, 289.2.a.d$^{3}$, 578.2.a.a$^{2}$, 578.2.a.b$^{2}$, 578.2.a.c$^{2}$, 578.2.a.d$^{2}$, 578.2.a.e$^{2}$, 578.2.a.h$^{2}$, 1156.2.a.a, 1156.2.a.c, 1156.2.a.d, 1156.2.a.f, 1156.2.a.h |
Rational points
This modular curve has 3 rational cusps and 1 rational CM point, but no other known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.12.0-2.a.1.2 | $68$ | $153$ | $153$ | $0$ | $0$ | full Jacobian |
| 68.612.22-34.a.1.1 | $68$ | $3$ | $3$ | $22$ | $6$ | $1^{8}\cdot2^{8}\cdot3^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.3672.127-34.a.1.4 | $68$ | $2$ | $2$ | $127$ | $43$ | $1^{5}\cdot2^{5}\cdot3^{8}\cdot4^{6}$ |
| 68.3672.127-34.b.1.4 | $68$ | $2$ | $2$ | $127$ | $56$ | $1^{5}\cdot2^{5}\cdot3^{8}\cdot4^{6}$ |
| 68.3672.127-68.a.1.3 | $68$ | $2$ | $2$ | $127$ | $87$ | $1^{5}\cdot2^{5}\cdot3^{8}\cdot4^{6}$ |
| 68.3672.127-68.b.1.4 | $68$ | $2$ | $2$ | $127$ | $52$ | $1^{5}\cdot2^{5}\cdot3^{8}\cdot4^{6}$ |
| 68.3672.136-68.a.1.7 | $68$ | $2$ | $2$ | $136$ | $62$ | $1^{10}\cdot2^{17}\cdot3^{6}\cdot4\cdot6$ |
| 68.3672.136-68.b.1.12 | $68$ | $2$ | $2$ | $136$ | $54$ | $1^{10}\cdot2^{17}\cdot3^{6}\cdot4\cdot6$ |
| 68.3672.136-68.c.1.8 | $68$ | $2$ | $2$ | $136$ | $62$ | $1^{10}\cdot2^{17}\cdot3^{6}\cdot4\cdot6$ |
| 68.3672.136-68.d.1.7 | $68$ | $2$ | $2$ | $136$ | $54$ | $1^{10}\cdot2^{17}\cdot3^{6}\cdot4\cdot6$ |
| 68.3672.136-68.e.1.8 | $68$ | $2$ | $2$ | $136$ | $59$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
| 68.3672.136-68.f.1.7 | $68$ | $2$ | $2$ | $136$ | $60$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
| 68.3672.136-68.g.1.7 | $68$ | $2$ | $2$ | $136$ | $57$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
| 68.3672.136-68.h.1.8 | $68$ | $2$ | $2$ | $136$ | $58$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |