Invariants
| Level: | $68$ | $\SL_2$-level: | $4$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot4$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $2$ 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: | 4C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.6.0.1 |
Level structure
| $\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}23&14\\65&35\end{bmatrix}$, $\begin{bmatrix}45&66\\64&3\end{bmatrix}$, $\begin{bmatrix}67&58\\39&29\end{bmatrix}$ |
| 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: | $1253376$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 770 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}\cdot17}\cdot\frac{x^{6}(17x^{2}-4096y^{2})^{3}}{y^{4}x^{8}}$ |
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 |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 68.12.0.a.1 | $68$ | $2$ | $2$ | $0$ |
| 68.12.0.d.1 | $68$ | $2$ | $2$ | $0$ |
| 68.12.0.e.1 | $68$ | $2$ | $2$ | $0$ |
| 68.12.0.h.1 | $68$ | $2$ | $2$ | $0$ |
| 68.108.7.m.1 | $68$ | $18$ | $18$ | $7$ |
| 68.816.57.h.1 | $68$ | $136$ | $136$ | $57$ |
| 68.918.64.h.1 | $68$ | $153$ | $153$ | $64$ |
| 136.12.0.f.1 | $136$ | $2$ | $2$ | $0$ |
| 136.12.0.l.1 | $136$ | $2$ | $2$ | $0$ |
| 136.12.0.o.1 | $136$ | $2$ | $2$ | $0$ |
| 136.12.0.x.1 | $136$ | $2$ | $2$ | $0$ |
| 204.12.0.r.1 | $204$ | $2$ | $2$ | $0$ |
| 204.12.0.t.1 | $204$ | $2$ | $2$ | $0$ |
| 204.12.0.v.1 | $204$ | $2$ | $2$ | $0$ |
| 204.12.0.x.1 | $204$ | $2$ | $2$ | $0$ |
| 204.18.0.l.1 | $204$ | $3$ | $3$ | $0$ |
| 204.24.1.p.1 | $204$ | $4$ | $4$ | $1$ |