Invariants
| Level: | $63$ | $\SL_2$-level: | $7$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $1^{3}\cdot7^{3}$ | Cusp orbits | $3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 7E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 63.48.0.11 |
Level structure
| $\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}22&38\\28&46\end{bmatrix}$, $\begin{bmatrix}47&11\\42&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 63.24.0.b.1 for the level structure with $-I$) |
| Cyclic 63-isogeny field degree: | $12$ |
| Cyclic 63-torsion field degree: | $432$ |
| Full 63-torsion field degree: | $163296$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -3^2\,\frac{(x+2y)^{24}(4x^{2}-2xy+7y^{2})^{3}(76864x^{6}+27552x^{5}y-940800x^{4}y^{2}+96040x^{3}y^{3}+2304960x^{2}y^{4}-705894xy^{5}+16807y^{6})^{3}}{(x+2y)^{24}(8x^{3}+12x^{2}y-48xy^{2}+y^{3})(40x^{3}+168x^{2}y-294xy^{2}-49y^{3})^{7}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 7.16.0-7.a.1.1 | $7$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 63.144.4-63.b.2.1 | $63$ | $3$ | $3$ | $4$ |
| 63.192.3-63.b.1.4 | $63$ | $4$ | $4$ | $3$ |
| 63.336.3-63.b.1.3 | $63$ | $7$ | $7$ | $3$ |
| 63.1296.46-63.c.2.1 | $63$ | $27$ | $27$ | $46$ |
| 126.96.2-126.e.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.96.2-126.i.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.96.2-126.j.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.96.2-126.m.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.144.1-126.l.1.6 | $126$ | $3$ | $3$ | $1$ |
| 252.96.2-252.l.2.10 | $252$ | $2$ | $2$ | $2$ |
| 252.96.2-252.p.2.6 | $252$ | $2$ | $2$ | $2$ |
| 252.96.2-252.q.2.7 | $252$ | $2$ | $2$ | $2$ |
| 252.96.2-252.t.2.6 | $252$ | $2$ | $2$ | $2$ |
| 252.192.6-252.bl.1.3 | $252$ | $4$ | $4$ | $6$ |
| 315.240.8-315.b.1.2 | $315$ | $5$ | $5$ | $8$ |
| 315.288.7-315.b.2.10 | $315$ | $6$ | $6$ | $7$ |
| 315.480.15-315.f.1.4 | $315$ | $10$ | $10$ | $15$ |