Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $4$ | $\PSL_2$-index: | $2$ | ||||
| Genus: | $0 = 1 + \frac{ 2 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $2$ | Cusp orbits | $1$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2A0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.4.0.1 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}10&1\\7&9\end{bmatrix}$, $\begin{bmatrix}31&29\\25&0\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 2.2.0.a.1 for the level structure with $-I$) |
| Cyclic 68-isogeny field degree: | $108$ |
| Cyclic 68-torsion field degree: | $3456$ |
| Full 68-torsion field degree: | $1880064$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 32740 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 2 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{2}(x^{2}+1728y^{2})}{y^{2}x^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.12.0-2.a.1.2 | $68$ | $3$ | $3$ | $0$ |
| 68.16.0-4.a.1.1 | $68$ | $4$ | $4$ | $0$ |
| 204.12.1-6.a.1.4 | $204$ | $3$ | $3$ | $1$ |
| 204.16.0-6.a.1.4 | $204$ | $4$ | $4$ | $0$ |
| 68.72.3-34.a.1.4 | $68$ | $18$ | $18$ | $3$ |
| 68.544.19-34.a.1.4 | $68$ | $136$ | $136$ | $19$ |
| 68.612.22-34.a.1.1 | $68$ | $153$ | $153$ | $22$ |