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: | 63.48.0.20 |
Level structure
| $\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}30&62\\56&5\end{bmatrix}$, $\begin{bmatrix}40&12\\14&20\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 63.24.0.c.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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^2\cdot7^5}{2}\cdot\frac{(x+2y)^{24}(4x^{2}-2xy+7y^{2})^{3}(1088x^{6}+672x^{5}y-188160x^{4}y^{2}+174440x^{3}y^{3}+164640x^{2}y^{4}-96726xy^{5}-43561y^{6})^{3}}{(x+2y)^{24}(40x^{3}+168x^{2}y-294xy^{2}-49y^{3})^{7}(64x^{3}+798x^{2}y-735xy^{2}-343y^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.16.0-7.a.1.1 | $21$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 63.144.4-63.c.2.6 | $63$ | $3$ | $3$ | $4$ |
| 63.192.3-63.c.2.6 | $63$ | $4$ | $4$ | $3$ |
| 63.336.3-63.a.1.2 | $63$ | $7$ | $7$ | $3$ |
| 63.1296.46-63.e.2.3 | $63$ | $27$ | $27$ | $46$ |
| 126.96.2-126.f.1.3 | $126$ | $2$ | $2$ | $2$ |
| 126.96.2-126.g.1.3 | $126$ | $2$ | $2$ | $2$ |
| 126.96.2-126.n.1.3 | $126$ | $2$ | $2$ | $2$ |
| 126.96.2-126.o.1.3 | $126$ | $2$ | $2$ | $2$ |
| 126.144.1-126.m.2.8 | $126$ | $3$ | $3$ | $1$ |
| 252.96.2-252.m.2.12 | $252$ | $2$ | $2$ | $2$ |
| 252.96.2-252.n.2.8 | $252$ | $2$ | $2$ | $2$ |
| 252.96.2-252.u.2.8 | $252$ | $2$ | $2$ | $2$ |
| 252.96.2-252.v.2.8 | $252$ | $2$ | $2$ | $2$ |
| 252.192.6-252.bm.2.16 | $252$ | $4$ | $4$ | $6$ |
| 315.240.8-315.c.2.6 | $315$ | $5$ | $5$ | $8$ |
| 315.288.7-315.k.1.10 | $315$ | $6$ | $6$ | $7$ |
| 315.480.15-315.g.1.16 | $315$ | $10$ | $10$ | $15$ |