Invariants
Level: | $68$ | $\SL_2$-level: | $4$ | ||||
Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
Cusps: | $2$ (none of which are rational) | Cusp widths | $4^{2}$ | Cusp orbits | $2$ | ||
Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4D0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 68.16.0.1 |
Level structure
$\GL_2(\Z/68\Z)$-generators: | $\begin{bmatrix}7&66\\14&41\end{bmatrix}$, $\begin{bmatrix}41&41\\59&12\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 4.8.0.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: | $470016$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ x^{2} + 16 x z + y^{2} - 16 y z + 144 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
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 | $68$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
68.48.0-4.a.1.1 | $68$ | $3$ | $3$ | $0$ |
68.288.11-68.a.1.1 | $68$ | $18$ | $18$ | $11$ |
68.2176.83-68.a.1.3 | $68$ | $136$ | $136$ | $83$ |
68.2448.94-68.a.1.3 | $68$ | $153$ | $153$ | $94$ |
136.64.1-8.a.1.2 | $136$ | $4$ | $4$ | $1$ |
204.48.2-12.a.1.4 | $204$ | $3$ | $3$ | $2$ |
204.64.1-12.a.1.4 | $204$ | $4$ | $4$ | $1$ |