Invariants
| Level: | $21$ | $\SL_2$-level: | $7$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $3$ are rational) | Cusp widths | $1^{3}\cdot7^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 7E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 21.48.0.1 |
Level structure
| $\GL_2(\Z/21\Z)$-generators: | $\begin{bmatrix}9&11\\5&8\end{bmatrix}$, $\begin{bmatrix}17&19\\12&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 7.24.0.a.1 for the level structure with $-I$) |
| Cyclic 21-isogeny field degree: | $4$ |
| Cyclic 21-torsion field degree: | $16$ |
| Full 21-torsion field degree: | $2016$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 80 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^7\cdot3^7}\cdot\frac{x^{24}(7x^{2}-30xy+36y^{2})^{3}(7x^{6}+1302x^{5}y-10080x^{4}y^{2}+22680x^{3}y^{3}-54432xy^{5}+46656y^{6})^{3}}{x^{31}(x-3y)^{7}(x-2y)^{7}(29x^{3}-96x^{2}y-36xy^{2}+216y^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.16.0-7.a.1.2 | $21$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 21.144.4-21.a.1.6 | $21$ | $3$ | $3$ | $4$ |
| 21.192.3-21.a.1.6 | $21$ | $4$ | $4$ | $3$ |
| 21.336.3-7.b.1.2 | $21$ | $7$ | $7$ | $3$ |
| 42.96.2-42.a.1.1 | $42$ | $2$ | $2$ | $2$ |
| 42.96.2-14.c.1.1 | $42$ | $2$ | $2$ | $2$ |
| 42.96.2-14.e.1.1 | $42$ | $2$ | $2$ | $2$ |
| 42.96.2-42.e.1.1 | $42$ | $2$ | $2$ | $2$ |
| 42.144.1-14.a.1.1 | $42$ | $3$ | $3$ | $1$ |
| 63.1296.46-63.a.2.3 | $63$ | $27$ | $27$ | $46$ |
| 84.96.2-28.d.1.4 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-28.f.1.4 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.g.1.1 | $84$ | $2$ | $2$ | $2$ |
| 84.96.2-84.p.1.1 | $84$ | $2$ | $2$ | $2$ |
| 84.192.6-28.k.2.5 | $84$ | $4$ | $4$ | $6$ |
| 105.240.8-35.a.2.3 | $105$ | $5$ | $5$ | $8$ |
| 105.288.7-35.a.2.2 | $105$ | $6$ | $6$ | $7$ |
| 105.480.15-35.a.2.4 | $105$ | $10$ | $10$ | $15$ |
| 147.336.3-49.b.1.2 | $147$ | $7$ | $7$ | $3$ |
| 168.96.2-56.f.1.7 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.g.1.7 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.j.1.7 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-56.k.1.7 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.l.1.10 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.m.1.11 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.bf.1.6 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.bg.1.7 | $168$ | $2$ | $2$ | $2$ |
| 210.96.2-70.a.1.3 | $210$ | $2$ | $2$ | $2$ |
| 210.96.2-210.a.1.3 | $210$ | $2$ | $2$ | $2$ |
| 210.96.2-70.d.1.3 | $210$ | $2$ | $2$ | $2$ |
| 210.96.2-210.d.1.3 | $210$ | $2$ | $2$ | $2$ |