Invariants
| Level: | $273$ | $\SL_2$-level: | $21$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $1^{3}\cdot3^{3}\cdot7^{3}\cdot21^{3}$ | Cusp orbits | $3^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 21D3 |
Level structure
| $\GL_2(\Z/273\Z)$-generators: | $\begin{bmatrix}3&154\\20&25\end{bmatrix}$, $\begin{bmatrix}82&140\\144&218\end{bmatrix}$, $\begin{bmatrix}152&7\\87&184\end{bmatrix}$, $\begin{bmatrix}264&127\\151&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 273.96.3.b.2 for the level structure with $-I$) |
| Cyclic 273-isogeny field degree: | $14$ |
| Cyclic 273-torsion field degree: | $2016$ |
| Full 273-torsion field degree: | $13208832$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 91.48.0-91.b.2.2 | $91$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.64.1-21.a.1.3 | $21$ | $3$ | $3$ | $1$ | $0$ |
| 91.48.0-91.b.2.2 | $91$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 273.384.5-273.h.2.3 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.h.3.8 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.i.1.4 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.i.4.7 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.l.2.4 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.l.4.6 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.m.2.2 | $273$ | $2$ | $2$ | $5$ |
| 273.384.5-273.m.4.8 | $273$ | $2$ | $2$ | $5$ |