Invariants
| Level: | $28$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 28.12.0.1 |
Level structure
| $\GL_2(\Z/28\Z)$-generators: | $\begin{bmatrix}3&14\\0&11\end{bmatrix}$, $\begin{bmatrix}3&14\\26&15\end{bmatrix}$, $\begin{bmatrix}13&6\\16&5\end{bmatrix}$, $\begin{bmatrix}17&8\\6&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 2.6.0.a.1 for the level structure with $-I$) |
| Cyclic 28-isogeny field degree: | $16$ |
| Cyclic 28-torsion field degree: | $192$ |
| Full 28-torsion field degree: | $16128$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 31720 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+192y^{2})^{3}}{y^{2}x^{6}(x-8y)^{2}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 28.24.0-4.a.1.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-4.a.1.2 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-4.b.1.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-4.b.1.2 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-4.b.1.3 | $28$ | $2$ | $2$ | $0$ |
| 84.36.1-6.a.1.1 | $84$ | $3$ | $3$ | $1$ |
| 84.48.0-6.a.1.2 | $84$ | $4$ | $4$ | $0$ |
| 56.24.0-8.a.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.a.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.a.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.b.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.b.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-8.b.1.3 | $56$ | $2$ | $2$ | $0$ |
| 140.60.2-10.a.1.1 | $140$ | $5$ | $5$ | $2$ |
| 140.72.1-10.a.1.1 | $140$ | $6$ | $6$ | $1$ |
| 140.120.3-10.a.1.3 | $140$ | $10$ | $10$ | $3$ |
| 84.24.0-12.a.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.a.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.a.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.b.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.b.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-12.b.1.3 | $84$ | $2$ | $2$ | $0$ |
| 28.96.2-14.a.1.3 | $28$ | $8$ | $8$ | $2$ |
| 28.252.7-14.a.1.2 | $28$ | $21$ | $21$ | $7$ |
| 28.336.9-14.a.1.3 | $28$ | $28$ | $28$ | $9$ |
| 252.324.10-18.a.1.1 | $252$ | $27$ | $27$ | $10$ |
| 140.24.0-20.a.1.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-20.a.1.2 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-20.a.1.3 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-20.b.1.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-20.b.1.2 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-20.b.1.3 | $140$ | $2$ | $2$ | $0$ |
| 308.144.4-22.a.1.2 | $308$ | $12$ | $12$ | $4$ |
| 168.24.0-24.a.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.a.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.a.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.b.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.b.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-24.b.1.6 | $168$ | $2$ | $2$ | $0$ |
| 28.24.0-28.a.1.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.a.1.2 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.a.1.4 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.b.1.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.b.1.2 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.b.1.4 | $28$ | $2$ | $2$ | $0$ |
| 280.24.0-40.a.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-40.a.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-40.a.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-40.b.1.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-40.b.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-40.b.1.5 | $280$ | $2$ | $2$ | $0$ |
| 308.24.0-44.a.1.1 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-44.a.1.2 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-44.a.1.3 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-44.b.1.1 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-44.b.1.2 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-44.b.1.3 | $308$ | $2$ | $2$ | $0$ |
| 56.24.0-56.a.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.a.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.a.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.b.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.b.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.b.1.5 | $56$ | $2$ | $2$ | $0$ |
| 84.24.0-84.a.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.a.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.a.1.5 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.3 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.6 | $84$ | $2$ | $2$ | $0$ |
| 140.24.0-140.a.1.2 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.a.1.3 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.a.1.7 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.b.1.3 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.b.1.4 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.b.1.7 | $140$ | $2$ | $2$ | $0$ |
| 168.24.0-168.a.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.a.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.a.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.11 | $168$ | $2$ | $2$ | $0$ |
| 280.24.0-280.a.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.a.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.a.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.b.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.b.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.b.1.11 | $280$ | $2$ | $2$ | $0$ |
| 308.24.0-308.a.1.2 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-308.a.1.3 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-308.a.1.8 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-308.b.1.3 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-308.b.1.4 | $308$ | $2$ | $2$ | $0$ |
| 308.24.0-308.b.1.8 | $308$ | $2$ | $2$ | $0$ |