Invariants
| Level: | $68$ | $\SL_2$-level: | $68$ | Newform level: | $4624$ | ||
| Index: | $1632$ | $\PSL_2$-index: | $1632$ | ||||
| Genus: | $121 = 1 + \frac{ 1632 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $34^{16}\cdot68^{16}$ | Cusp orbits | $8^{2}\cdot16$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $58$ | ||||||
| $\Q$-gonality: | $28 \le \gamma \le 72$ | ||||||
| $\overline{\Q}$-gonality: | $28 \le \gamma \le 72$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.1632.121.23 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}3&64\\10&55\end{bmatrix}$, $\begin{bmatrix}5&66\\66&29\end{bmatrix}$, $\begin{bmatrix}45&30\\34&23\end{bmatrix}$, $\begin{bmatrix}55&18\\6&25\end{bmatrix}$, $\begin{bmatrix}65&0\\56&37\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 68.3264.121-68.c.1.1, 68.3264.121-68.c.1.2, 68.3264.121-68.c.1.3, 68.3264.121-68.c.1.4, 68.3264.121-68.c.1.5, 68.3264.121-68.c.1.6, 68.3264.121-68.c.1.7, 68.3264.121-68.c.1.8 |
| Cyclic 68-isogeny field degree: | $36$ |
| Cyclic 68-torsion field degree: | $1152$ |
| Full 68-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{308}\cdot17^{234}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{27}\cdot3^{14}\cdot4^{2}\cdot6$ |
| Newforms: | 272.2.a.a, 272.2.a.b, 272.2.a.c, 272.2.a.d, 272.2.a.e, 272.2.a.f, 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, 4624.2.a.b, 4624.2.a.ba, 4624.2.a.bd, 4624.2.a.bf, 4624.2.a.bh, 4624.2.a.bi, 4624.2.a.bk, 4624.2.a.bq, 4624.2.a.bs, 4624.2.a.e, 4624.2.a.i, 4624.2.a.j, 4624.2.a.l, 4624.2.a.m, 4624.2.a.n, 4624.2.a.o, 4624.2.a.p, 4624.2.a.q, 4624.2.a.r, 4624.2.a.s, 4624.2.a.t, 4624.2.a.u, 4624.2.a.y |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5$, 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 |
|---|---|---|---|---|---|---|
| 34.816.57.a.1 | $34$ | $2$ | $2$ | $57$ | $28$ | $1^{6}\cdot2^{15}\cdot3^{6}\cdot4\cdot6$ |
| 68.12.0.a.1 | $68$ | $136$ | $136$ | $0$ | $0$ | full Jacobian |
| 68.816.57.e.1 | $68$ | $2$ | $2$ | $57$ | $34$ | $1^{6}\cdot2^{15}\cdot3^{6}\cdot4\cdot6$ |
| 68.816.57.h.1 | $68$ | $2$ | $2$ | $57$ | $30$ | $1^{4}\cdot2^{14}\cdot3^{8}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.3264.241.e.1 | $68$ | $2$ | $2$ | $241$ | $101$ | $2^{8}\cdot3^{14}\cdot4^{11}\cdot6\cdot12$ |
| 68.3264.241.g.1 | $68$ | $2$ | $2$ | $241$ | $142$ | $2^{8}\cdot3^{14}\cdot4^{11}\cdot6\cdot12$ |
| 68.3264.241.m.1 | $68$ | $2$ | $2$ | $241$ | $111$ | $2^{8}\cdot3^{14}\cdot4^{11}\cdot6\cdot12$ |
| 68.3264.241.o.1 | $68$ | $2$ | $2$ | $241$ | $110$ | $2^{8}\cdot3^{14}\cdot4^{11}\cdot6\cdot12$ |
| 68.3264.249.c.1 | $68$ | $2$ | $2$ | $249$ | $120$ | $1^{12}\cdot2^{30}\cdot3^{12}\cdot4^{2}\cdot6^{2}$ |
| 68.3264.249.e.1 | $68$ | $2$ | $2$ | $249$ | $106$ | $1^{12}\cdot2^{30}\cdot3^{12}\cdot4^{2}\cdot6^{2}$ |
| 68.3264.249.l.1 | $68$ | $2$ | $2$ | $249$ | $114$ | $2^{8}\cdot3^{12}\cdot4^{10}\cdot6^{2}\cdot12^{2}$ |
| 68.3264.249.m.1 | $68$ | $2$ | $2$ | $249$ | $114$ | $2^{8}\cdot3^{12}\cdot4^{10}\cdot6^{2}\cdot12^{2}$ |
| 68.4896.361.c.1 | $68$ | $3$ | $3$ | $361$ | $157$ | $1^{20}\cdot2^{30}\cdot3^{28}\cdot4^{13}\cdot6^{2}\cdot12$ |