Invariants
| Level: | $63$ | $\SL_2$-level: | $9$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $1^{3}\cdot3^{2}\cdot9^{3}$ | Cusp orbits | $2\cdot3^{2}$ | ||
| 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: | 9I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 63.72.0.8 |
Level structure
| $\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}4&54\\38&40\end{bmatrix}$, $\begin{bmatrix}11&54\\33&10\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 63.36.0.e.1 for the level structure with $-I$) |
| Cyclic 63-isogeny field degree: | $8$ |
| Cyclic 63-torsion field degree: | $144$ |
| Full 63-torsion field degree: | $108864$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 10 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{7}\cdot\frac{x^{36}(2x^{3}+9x^{2}y+3xy^{2}-2y^{3})^{3}(32x^{9}+180x^{8}y+450x^{7}y^{2}+693x^{6}y^{3}+567x^{5}y^{4}+189x^{4}y^{5}-105x^{3}y^{6}-162x^{2}y^{7}-108xy^{8}-32y^{9})^{3}}{x^{36}(x^{2}+xy+y^{2})^{3}(x^{3}+x^{2}y-2xy^{2}-y^{3})^{9}(5x^{3}+12x^{2}y-3xy^{2}-5y^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 9.24.0-9.a.1.2 | $9$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 63.216.1-63.b.2.3 | $63$ | $3$ | $3$ | $1$ |
| 63.576.17-63.n.1.1 | $63$ | $8$ | $8$ | $17$ |
| 63.1512.52-63.i.2.1 | $63$ | $21$ | $21$ | $52$ |
| 63.2016.69-63.i.2.1 | $63$ | $28$ | $28$ | $69$ |
| 126.144.3-126.i.2.1 | $126$ | $2$ | $2$ | $3$ |
| 126.144.3-126.ba.2.1 | $126$ | $2$ | $2$ | $3$ |
| 126.144.3-126.bl.2.1 | $126$ | $2$ | $2$ | $3$ |
| 126.144.3-126.bp.2.1 | $126$ | $2$ | $2$ | $3$ |
| 126.216.2-126.e.2.4 | $126$ | $3$ | $3$ | $2$ |
| 189.216.1-189.b.1.2 | $189$ | $3$ | $3$ | $1$ |
| 189.216.4-189.b.1.2 | $189$ | $3$ | $3$ | $4$ |
| 189.216.4-189.e.2.4 | $189$ | $3$ | $3$ | $4$ |
| 252.144.3-252.e.1.9 | $252$ | $2$ | $2$ | $3$ |
| 252.144.3-252.bc.1.5 | $252$ | $2$ | $2$ | $3$ |
| 252.144.3-252.bv.1.5 | $252$ | $2$ | $2$ | $3$ |
| 252.144.3-252.bz.1.5 | $252$ | $2$ | $2$ | $3$ |
| 252.288.9-252.hy.2.9 | $252$ | $4$ | $4$ | $9$ |
| 315.360.12-315.e.1.1 | $315$ | $5$ | $5$ | $12$ |
| 315.432.11-315.e.2.5 | $315$ | $6$ | $6$ | $11$ |