Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.24.0.7 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}7&30\\20&51\end{bmatrix}$, $\begin{bmatrix}39&0\\58&3\end{bmatrix}$, $\begin{bmatrix}39&66\\26&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 68.12.0.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: | $313344$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 178 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}\cdot17^2}\cdot\frac{x^{12}(289x^{4}+69632x^{2}y^{2}+16777216y^{4})^{3}}{y^{4}x^{16}(17x^{2}+4096y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 4.12.0-2.a.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
| 68.12.0-2.a.1.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.48.0-68.a.1.2 | $68$ | $2$ | $2$ | $0$ |
| 68.48.0-68.c.1.3 | $68$ | $2$ | $2$ | $0$ |
| 68.48.0-68.c.1.4 | $68$ | $2$ | $2$ | $0$ |
| 68.432.15-68.c.1.5 | $68$ | $18$ | $18$ | $15$ |
| 68.3264.121-68.c.1.8 | $68$ | $136$ | $136$ | $121$ |
| 68.3672.136-68.c.1.7 | $68$ | $153$ | $153$ | $136$ |
| 136.48.0-136.b.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.b.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.f.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.48.0-136.f.1.6 | $136$ | $2$ | $2$ | $0$ |
| 204.48.0-204.d.1.3 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.d.1.6 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.f.1.3 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.f.1.6 | $204$ | $2$ | $2$ | $0$ |
| 204.72.2-204.a.1.6 | $204$ | $3$ | $3$ | $2$ |
| 204.96.1-204.a.1.8 | $204$ | $4$ | $4$ | $1$ |