Invariants
| Level: | $204$ | $\SL_2$-level: | $68$ | Newform level: | $1$ | ||
| Index: | $216$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $15 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot34^{2}\cdot68^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 15$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 15$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 68D15 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}21&136\\50&29\end{bmatrix}$, $\begin{bmatrix}25&0\\113&103\end{bmatrix}$, $\begin{bmatrix}31&136\\183&1\end{bmatrix}$, $\begin{bmatrix}87&68\\1&109\end{bmatrix}$, $\begin{bmatrix}125&136\\142&75\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 204.432.15-204.p.1.1, 204.432.15-204.p.1.2, 204.432.15-204.p.1.3, 204.432.15-204.p.1.4, 204.432.15-204.p.1.5, 204.432.15-204.p.1.6, 204.432.15-204.p.1.7, 204.432.15-204.p.1.8 |
| Cyclic 204-isogeny field degree: | $4$ |
| Cyclic 204-torsion field degree: | $256$ |
| Full 204-torsion field degree: | $1671168$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0.h.1 | $12$ | $18$ | $18$ | $0$ | $0$ |
| $X_0(17)$ | $17$ | $12$ | $12$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0.h.1 | $12$ | $18$ | $18$ | $0$ | $0$ |
| $X_0(68)$ | $68$ | $2$ | $2$ | $7$ | $0$ |
| 204.108.7.a.1 | $204$ | $2$ | $2$ | $7$ | $?$ |
| 204.108.7.q.1 | $204$ | $2$ | $2$ | $7$ | $?$ |