Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.24.0.17 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}53&34\\50&59\end{bmatrix}$, $\begin{bmatrix}55&19\\12&33\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 68-isogeny field degree: | $36$ |
| Cyclic 68-torsion field degree: | $1152$ |
| Full 68-torsion field degree: | $313344$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 9 x^{2} + 16 x y + 128 y^{2} + 17 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.12.0.e.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 68.8.0.a.1 | $68$ | $3$ | $3$ | $0$ | $0$ |
| 68.12.0.k.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 68.12.0.m.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.432.31.w.1 | $68$ | $18$ | $18$ | $31$ |
| 68.3264.249.x.1 | $68$ | $136$ | $136$ | $249$ |
| 68.3672.280.bb.1 | $68$ | $153$ | $153$ | $280$ |
| 136.48.1.eu.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.fe.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.ge.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.gk.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.hk.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.hq.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.iw.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.jg.1 | $136$ | $2$ | $2$ | $1$ |
| 204.72.4.cg.1 | $204$ | $3$ | $3$ | $4$ |
| 204.96.3.bs.1 | $204$ | $4$ | $4$ | $3$ |