Invariants
| Level: | $63$ | $\SL_2$-level: | $9$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{3}\cdot9^{3}$ | Cusp orbits | $3^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ 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: | 9J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 63.72.0.15 |
Level structure
| $\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}20&30\\49&19\end{bmatrix}$, $\begin{bmatrix}29&27\\9&10\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 63.36.0.i.1 for the level structure with $-I$) |
| Cyclic 63-isogeny field degree: | $24$ |
| Cyclic 63-torsion field degree: | $432$ |
| 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3\cdot7}{2^9}\cdot\frac{(7x+24y)^{36}(7x^{2}-6xy+36y^{2})^{3}(91x^{3}+126x^{2}y-1512xy^{2}+216y^{3})(561491x^{9}+7149492x^{8}y+30561300x^{7}y^{2}-189114912x^{6}y^{3}-817296480x^{5}y^{4}+3521804832x^{4}y^{5}-1776333888x^{3}y^{6}-2345583744x^{2}y^{7}-1375605504xy^{8}-10077696y^{9})^{3}}{(7x+24y)^{36}(28x^{3}-63x^{2}y-378xy^{2}+216y^{3})^{9}(35x^{3}+252x^{2}y-756xy^{2}-216y^{3})^{3}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 9.24.0-9.b.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.c.1.3 | $63$ | $3$ | $3$ | $1$ |
| 63.576.17-63.bd.2.8 | $63$ | $8$ | $8$ | $17$ |
| 63.1512.55-63.j.2.8 | $63$ | $21$ | $21$ | $55$ |
| 63.2016.72-63.d.2.8 | $63$ | $28$ | $28$ | $72$ |
| 126.144.2-126.f.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.144.2-126.i.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.144.2-126.l.1.2 | $126$ | $2$ | $2$ | $2$ |
| 126.144.2-126.r.2.3 | $126$ | $2$ | $2$ | $2$ |
| 126.216.4-126.i.1.4 | $126$ | $3$ | $3$ | $4$ |
| 189.216.4-189.i.2.3 | $189$ | $3$ | $3$ | $4$ |
| 189.216.7-189.c.2.3 | $189$ | $3$ | $3$ | $7$ |
| 189.216.7-189.f.2.4 | $189$ | $3$ | $3$ | $7$ |
| 252.144.2-252.c.1.9 | $252$ | $2$ | $2$ | $2$ |
| 252.144.2-252.f.1.5 | $252$ | $2$ | $2$ | $2$ |
| 252.144.2-252.i.1.5 | $252$ | $2$ | $2$ | $2$ |
| 252.144.2-252.o.1.5 | $252$ | $2$ | $2$ | $2$ |
| 252.288.9-252.is.1.4 | $252$ | $4$ | $4$ | $9$ |
| 315.360.11-315.c.1.8 | $315$ | $5$ | $5$ | $11$ |
| 315.432.13-315.f.1.12 | $315$ | $6$ | $6$ | $13$ |