Invariants
| Level: | $21$ | $\SL_2$-level: | $7$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot7$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-7,-28$) |
Other labels
| Cummins and Pauli (CP) label: | 7B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 21.16.0.2 |
Level structure
| $\GL_2(\Z/21\Z)$-generators: | $\begin{bmatrix}17&19\\19&3\end{bmatrix}$, $\begin{bmatrix}19&7\\11&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 7.8.0.a.1 for the level structure with $-I$) |
| Cyclic 21-isogeny field degree: | $4$ |
| Cyclic 21-torsion field degree: | $48$ |
| Full 21-torsion field degree: | $6048$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 444 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2}\cdot\frac{x^{8}(x^{2}-6xy-12y^{2})^{3}(13x^{2}-30xy+36y^{2})}{x^{15}(2x+3y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 21.48.0-7.a.1.1 | $21$ | $3$ | $3$ | $0$ |
| 21.48.0-7.a.2.2 | $21$ | $3$ | $3$ | $0$ |
| 21.48.0-7.b.1.1 | $21$ | $3$ | $3$ | $0$ |
| 21.112.1-7.a.1.1 | $21$ | $7$ | $7$ | $1$ |
| 42.32.0-14.a.1.1 | $42$ | $2$ | $2$ | $0$ |
| 42.32.0-14.b.1.1 | $42$ | $2$ | $2$ | $0$ |
| 42.48.1-14.a.1.4 | $42$ | $3$ | $3$ | $1$ |
| 21.48.2-21.a.1.4 | $21$ | $3$ | $3$ | $2$ |
| 21.64.1-21.a.1.2 | $21$ | $4$ | $4$ | $1$ |
| 84.32.0-28.a.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.32.0-28.b.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.64.2-28.a.1.7 | $84$ | $4$ | $4$ | $2$ |
| 105.80.2-35.a.1.2 | $105$ | $5$ | $5$ | $2$ |
| 105.96.3-35.a.1.8 | $105$ | $6$ | $6$ | $3$ |
| 105.160.5-35.a.1.4 | $105$ | $10$ | $10$ | $5$ |
| 42.32.0-42.a.1.1 | $42$ | $2$ | $2$ | $0$ |
| 42.32.0-42.b.1.1 | $42$ | $2$ | $2$ | $0$ |
| 147.112.1-49.a.1.1 | $147$ | $7$ | $7$ | $1$ |
| 168.32.0-56.a.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.32.0-56.b.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.32.0-56.c.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.32.0-56.d.1.2 | $168$ | $2$ | $2$ | $0$ |
| 63.48.0-63.a.1.2 | $63$ | $3$ | $3$ | $0$ |
| 63.48.0-63.a.2.2 | $63$ | $3$ | $3$ | $0$ |
| 63.48.0-63.b.1.3 | $63$ | $3$ | $3$ | $0$ |
| 63.48.0-63.b.2.1 | $63$ | $3$ | $3$ | $0$ |
| 63.48.0-63.c.1.3 | $63$ | $3$ | $3$ | $0$ |
| 63.48.0-63.c.2.1 | $63$ | $3$ | $3$ | $0$ |
| 63.432.14-63.a.1.4 | $63$ | $27$ | $27$ | $14$ |
| 210.32.0-70.a.1.1 | $210$ | $2$ | $2$ | $0$ |
| 210.32.0-70.b.1.1 | $210$ | $2$ | $2$ | $0$ |
| 231.192.7-77.a.1.6 | $231$ | $12$ | $12$ | $7$ |
| 84.32.0-84.c.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.32.0-84.d.1.2 | $84$ | $2$ | $2$ | $0$ |
| 273.48.0-91.a.1.2 | $273$ | $3$ | $3$ | $0$ |
| 273.48.0-91.a.2.2 | $273$ | $3$ | $3$ | $0$ |
| 273.48.0-91.b.1.2 | $273$ | $3$ | $3$ | $0$ |
| 273.48.0-91.b.2.2 | $273$ | $3$ | $3$ | $0$ |
| 273.48.0-91.c.1.2 | $273$ | $3$ | $3$ | $0$ |
| 273.48.0-91.c.2.2 | $273$ | $3$ | $3$ | $0$ |
| 273.224.7-91.a.1.3 | $273$ | $14$ | $14$ | $7$ |
| 168.32.0-168.c.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.32.0-168.d.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.32.0-168.e.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.32.0-168.f.1.2 | $168$ | $2$ | $2$ | $0$ |
| 210.32.0-210.a.1.1 | $210$ | $2$ | $2$ | $0$ |
| 210.32.0-210.b.1.1 | $210$ | $2$ | $2$ | $0$ |