Invariants
| Level: | $28$ | $\SL_2$-level: | $4$ | ||||
| Index: | $4$ | $\PSL_2$-index: | $2$ | ||||
| Genus: | $0 = 1 + \frac{ 2 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $2$ | Cusp orbits | $1$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2A0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.4.0.2 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}11&20\\5&9\end{bmatrix}$, $\begin{bmatrix}25&17\\10&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.2.0.a.1 for the level structure with $-I$) |
| Cyclic 28-isogeny field degree: | $48$ |
| Cyclic 28-torsion field degree: | $576$ |
| Full 28-torsion field degree: | $48384$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 13525 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 2 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{2}(1728x^{2}-y^{2})}{x^{4}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 28.12.0-4.a.1.2 | $28$ | $3$ | $3$ | $0$ |
| 28.16.0-4.b.1.1 | $28$ | $4$ | $4$ | $0$ |
| 28.32.0-28.a.1.1 | $28$ | $8$ | $8$ | $0$ |
| 28.84.3-28.a.1.1 | $28$ | $21$ | $21$ | $3$ |
| 28.112.3-28.a.1.2 | $28$ | $28$ | $28$ | $3$ |
| 84.12.1-12.a.1.2 | $84$ | $3$ | $3$ | $1$ |
| 84.16.0-12.a.1.6 | $84$ | $4$ | $4$ | $0$ |
| 140.20.0-20.a.1.1 | $140$ | $5$ | $5$ | $0$ |
| 140.24.1-20.a.1.1 | $140$ | $6$ | $6$ | $1$ |
| 140.40.1-20.a.1.3 | $140$ | $10$ | $10$ | $1$ |
| 252.108.2-36.a.1.2 | $252$ | $27$ | $27$ | $2$ |
| 308.48.2-44.a.1.5 | $308$ | $12$ | $12$ | $2$ |
| 308.220.5-44.a.1.2 | $308$ | $55$ | $55$ | $5$ |
| 308.220.7-44.a.1.2 | $308$ | $55$ | $55$ | $7$ |
| 308.264.9-44.a.1.3 | $308$ | $66$ | $66$ | $9$ |