Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.12.0.2 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}25&2\\6&1\end{bmatrix}$, $\begin{bmatrix}55&2\\52&47\end{bmatrix}$, $\begin{bmatrix}55&28\\8&1\end{bmatrix}$, $\begin{bmatrix}55&34\\56&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 2.6.0.a.1 for the level structure with $-I$) |
| Cyclic 68-isogeny field degree: | $36$ |
| Cyclic 68-torsion field degree: | $1152$ |
| Full 68-torsion field degree: | $626688$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 31720 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+192y^{2})^{3}}{y^{2}x^{6}(x-8y)^{2}(x+8y)^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 68.4.0-2.a.1.1 | $68$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.24.0-4.a.1.2 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-4.b.1.2 | $68$ | $2$ | $2$ | $0$ |
| 204.36.1-6.a.1.5 | $204$ | $3$ | $3$ | $1$ |
| 204.48.0-6.a.1.9 | $204$ | $4$ | $4$ | $0$ |
| 136.24.0-8.a.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-8.b.1.4 | $136$ | $2$ | $2$ | $0$ |
| 204.24.0-12.a.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-12.b.1.4 | $204$ | $2$ | $2$ | $0$ |
| 68.216.7-34.a.1.6 | $68$ | $18$ | $18$ | $7$ |
| 68.1632.57-34.a.1.6 | $68$ | $136$ | $136$ | $57$ |
| 68.1836.64-34.a.1.5 | $68$ | $153$ | $153$ | $64$ |
| 68.24.0-68.a.1.3 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-68.b.1.3 | $68$ | $2$ | $2$ | $0$ |
| 136.24.0-136.a.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.b.1.6 | $136$ | $2$ | $2$ | $0$ |
| 204.24.0-204.a.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.b.1.8 | $204$ | $2$ | $2$ | $0$ |