Invariants
| Level: | $68$ | $\SL_2$-level: | $68$ | Newform level: | $1156$ | ||
| Index: | $3672$ | $\PSL_2$-index: | $1836$ | ||||
| Genus: | $127 = 1 + \frac{ 1836 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 54 }{2}$ | ||||||
| Cusps: | $54$ (none of which are rational) | Cusp widths | $34^{54}$ | Cusp orbits | $2^{3}\cdot16^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $56$ | ||||||
| $\Q$-gonality: | $33 \le \gamma \le 84$ | ||||||
| $\overline{\Q}$-gonality: | $33 \le \gamma \le 84$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.3672.127.10 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}11&26\\56&25\end{bmatrix}$, $\begin{bmatrix}13&4\\18&55\end{bmatrix}$, $\begin{bmatrix}31&12\\18&27\end{bmatrix}$, $\begin{bmatrix}39&62\\8&7\end{bmatrix}$, $\begin{bmatrix}51&16\\44&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 34.1836.127.b.1 for the level structure with $-I$) |
| Cyclic 68-isogeny field degree: | $4$ |
| Cyclic 68-torsion field degree: | $128$ |
| Full 68-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{102}\cdot17^{247}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{18}\cdot3^{16}\cdot4^{7}$ |
| Newforms: | 17.2.a.a$^{3}$, 34.2.a.a$^{2}$, 68.2.a.a, 289.2.a.a$^{6}$, 289.2.a.b$^{6}$, 289.2.a.d$^{6}$, 289.2.a.f$^{3}$, 578.2.a.a$^{4}$, 578.2.a.b$^{2}$, 578.2.a.c$^{2}$, 578.2.a.d$^{2}$, 578.2.a.e$^{4}$, 578.2.a.h$^{4}$, 578.2.a.i$^{2}$, 1156.2.a.a$^{2}$, 1156.2.a.c, 1156.2.a.d$^{2}$, 1156.2.a.f$^{2}$, 1156.2.a.g, 1156.2.a.h |
Rational points
This modular curve has no $\Q_p$ points for $p=5,29,37,\ldots,277$, 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 |
|---|---|---|---|---|---|---|
| 68.1224.43-34.b.1.1 | $68$ | $3$ | $3$ | $43$ | $16$ | $1^{12}\cdot2^{10}\cdot3^{12}\cdot4^{4}$ |
| 68.1836.64-34.a.1.5 | $68$ | $2$ | $2$ | $64$ | $28$ | $1^{5}\cdot2^{5}\cdot3^{8}\cdot4^{6}$ |
| 68.1836.64-34.a.1.7 | $68$ | $2$ | $2$ | $64$ | $28$ | $1^{5}\cdot2^{5}\cdot3^{8}\cdot4^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.7344.253-34.b.1.4 | $68$ | $2$ | $2$ | $253$ | $99$ | $1^{10}\cdot2^{20}\cdot3^{16}\cdot4^{7}$ |
| 68.7344.253-68.b.1.3 | $68$ | $2$ | $2$ | $253$ | $115$ | $1^{10}\cdot2^{20}\cdot3^{16}\cdot4^{7}$ |
| 68.7344.271-68.i.1.4 | $68$ | $2$ | $2$ | $271$ | $122$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.271-68.j.1.7 | $68$ | $2$ | $2$ | $271$ | $113$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.271-68.m.1.5 | $68$ | $2$ | $2$ | $271$ | $120$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.271-68.n.1.4 | $68$ | $2$ | $2$ | $271$ | $111$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.271-68.q.1.7 | $68$ | $2$ | $2$ | $271$ | $122$ | $1^{12}\cdot2^{24}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.271-68.r.1.7 | $68$ | $2$ | $2$ | $271$ | $112$ | $1^{12}\cdot2^{24}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |