Invariants
| Level: | $68$ | $\SL_2$-level: | $68$ | Newform level: | $4624$ | ||
| Index: | $3672$ | $\PSL_2$-index: | $1836$ | ||||
| Genus: | $136 = 1 + \frac{ 1836 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (of which $2$ are rational) | Cusp widths | $34^{18}\cdot68^{18}$ | Cusp orbits | $1^{2}\cdot2\cdot8^{2}\cdot16$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $58$ | ||||||
| $\Q$-gonality: | $32 \le \gamma \le 84$ | ||||||
| $\overline{\Q}$-gonality: | $32 \le \gamma \le 84$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.3672.136.89 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}23&14\\30&11\end{bmatrix}$, $\begin{bmatrix}27&52\\12&43\end{bmatrix}$, $\begin{bmatrix}33&48\\66&19\end{bmatrix}$, $\begin{bmatrix}47&52\\60&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 68.1836.136.h.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: | $2^{346}\cdot17^{257}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{14}\cdot2^{19}\cdot3^{14}\cdot4^{6}\cdot6\cdot12$ |
| Newforms: | 17.2.a.a$^{3}$, 34.2.a.a$^{2}$, 68.2.a.a, 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.ba, 4624.2.a.bd, 4624.2.a.bf, 4624.2.a.bh, 4624.2.a.bi, 4624.2.a.bk, 4624.2.a.bl, 4624.2.a.bm, 4624.2.a.bn, 4624.2.a.bo, 4624.2.a.bp, 4624.2.a.bs, 4624.2.a.bt, 4624.2.a.i, 4624.2.a.j, 4624.2.a.u, 4624.2.a.y |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.1836.64-34.a.1.4 | $68$ | $2$ | $2$ | $64$ | $28$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
| 68.1836.64-34.a.1.5 | $68$ | $2$ | $2$ | $64$ | $28$ | $1^{4}\cdot2^{6}\cdot3^{6}\cdot4^{5}\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 68.7344.271-68.a.1.4 | $68$ | $2$ | $2$ | $271$ | $107$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
| 68.7344.271-68.d.1.5 | $68$ | $2$ | $2$ | $271$ | $143$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
| 68.7344.271-68.m.1.5 | $68$ | $2$ | $2$ | $271$ | $120$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
| 68.7344.271-68.p.1.4 | $68$ | $2$ | $2$ | $271$ | $108$ | $1^{15}\cdot2^{22}\cdot3^{14}\cdot4^{7}\cdot6$ |
| 68.7344.280-68.j.1.4 | $68$ | $2$ | $2$ | $280$ | $122$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.280-68.k.1.4 | $68$ | $2$ | $2$ | $280$ | $113$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.280-68.k.1.7 | $68$ | $2$ | $2$ | $280$ | $113$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.280-68.m.1.5 | $68$ | $2$ | $2$ | $280$ | $124$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.280-68.m.1.7 | $68$ | $2$ | $2$ | $280$ | $124$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.280-68.o.1.3 | $68$ | $2$ | $2$ | $280$ | $115$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |
| 68.7344.280-68.o.1.8 | $68$ | $2$ | $2$ | $280$ | $115$ | $1^{14}\cdot2^{23}\cdot3^{12}\cdot4^{6}\cdot6^{2}\cdot12$ |