Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.48.0.11 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}51&46\\8&9\end{bmatrix}$, $\begin{bmatrix}57&2\\40&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.b.1 for the level structure with $-I$) |
| Cyclic 68-isogeny field degree: | $18$ |
| Cyclic 68-torsion field degree: | $576$ |
| Full 68-torsion field degree: | $156672$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 68.24.0-4.a.1.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 68.24.0-4.a.1.2 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 68.24.0-4.b.1.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 68.24.0-4.b.1.2 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 68.24.0-4.b.1.3 | $68$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.864.31-68.b.1.2 | $68$ | $18$ | $18$ | $31$ |
| 68.6528.249-68.b.1.2 | $68$ | $136$ | $136$ | $249$ |
| 68.7344.280-68.b.1.1 | $68$ | $153$ | $153$ | $280$ |
| 136.96.0-8.a.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.a.1.9 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.a.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.a.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.b.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.b.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.b.2.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.b.2.9 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.2 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.2.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.2.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.c.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-8.c.1.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.c.1.6 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.c.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-8.g.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.g.1.12 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.g.2.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.h.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.h.1.12 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-8.h.2.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.2.5 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.n.2.16 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.2.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.o.2.16 | $136$ | $2$ | $2$ | $1$ |
| 136.96.2-8.a.1.2 | $136$ | $2$ | $2$ | $2$ |
| 136.96.2-8.a.1.12 | $136$ | $2$ | $2$ | $2$ |
| 136.96.2-136.a.1.6 | $136$ | $2$ | $2$ | $2$ |
| 136.96.2-136.a.1.24 | $136$ | $2$ | $2$ | $2$ |
| 204.144.4-12.b.1.1 | $204$ | $3$ | $3$ | $4$ |
| 204.192.3-12.b.1.1 | $204$ | $4$ | $4$ | $3$ |