Invariants
| Level: | $273$ | $\SL_2$-level: | $21$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $1^{6}\cdot3^{6}\cdot7^{6}\cdot21^{6}$ | Cusp orbits | $3^{2}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 21E5 |
Level structure
| $\GL_2(\Z/273\Z)$-generators: | $\begin{bmatrix}64&81\\255&91\end{bmatrix}$, $\begin{bmatrix}95&244\\152&255\end{bmatrix}$, $\begin{bmatrix}195&163\\241&210\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 273.192.5.l.4 for the level structure with $-I$) |
| Cyclic 273-isogeny field degree: | $14$ |
| Cyclic 273-torsion field degree: | $2016$ |
| Full 273-torsion field degree: | $6604416$ |
Rational points
This modular curve has no $\Q_p$ points for $p=2$, and therefore no 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 |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $48$ | $48$ | $0$ | $0$ |
| 91.48.0-91.b.2.2 | $91$ | $8$ | $8$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.128.1-21.a.4.3 | $21$ | $3$ | $3$ | $1$ | $0$ |
| 273.192.3-273.b.2.6 | $273$ | $2$ | $2$ | $3$ | $?$ |
| 273.192.3-273.b.2.13 | $273$ | $2$ | $2$ | $3$ | $?$ |