Invariants
| Level: | $63$ | $\SL_2$-level: | $63$ | Newform level: | $189$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 12 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $3^{2}\cdot9^{2}\cdot21^{2}\cdot63^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $12$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 63F9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 63.384.9.28 |
Level structure
| $\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}34&33\\0&26\end{bmatrix}$, $\begin{bmatrix}40&39\\0&5\end{bmatrix}$, $\begin{bmatrix}41&1\\42&23\end{bmatrix}$, $\begin{bmatrix}43&31\\42&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 63.192.9.b.3 for the level structure with $-I$) |
| Cyclic 63-isogeny field degree: | $3$ |
| Cyclic 63-torsion field degree: | $108$ |
| Full 63-torsion field degree: | $20412$ |
Jacobian
| Conductor: | $3^{25}\cdot7^{9}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{5}\cdot4$ |
| Newforms: | 21.2.a.a, 189.2.a.a, 189.2.a.b, 189.2.a.c, 189.2.a.d, 189.2.c.c |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ y w + u v + v^{2} - v s $ |
| $=$ | $x y - y w - y t - u^{2} - u v - v r$ | |
| $=$ | $y z + y w - y t - u^{2} - u v + u s$ | |
| $=$ | $x y + y z + y w + y t + u v - u r$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.128.1-21.a.2.4 | $21$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot4$ |
| 63.192.5-63.b.1.6 | $63$ | $2$ | $2$ | $5$ | $2$ | $4$ |
| 63.192.5-63.b.1.7 | $63$ | $2$ | $2$ | $5$ | $2$ | $4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 63.1152.33-63.cd.4.2 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.ce.2.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.cf.4.2 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.cg.4.2 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.ch.4.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.ci.4.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.cj.3.3 | $63$ | $3$ | $3$ | $33$ | $6$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.ck.1.3 | $63$ | $3$ | $3$ | $33$ | $4$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cl.3.1 | $63$ | $3$ | $3$ | $33$ | $4$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cm.1.1 | $63$ | $3$ | $3$ | $33$ | $4$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cn.1.4 | $63$ | $3$ | $3$ | $33$ | $14$ | $1^{2}\cdot2\cdot5^{2}\cdot10$ |
| 63.1152.33-63.co.3.4 | $63$ | $3$ | $3$ | $33$ | $14$ | $1^{2}\cdot2\cdot5^{2}\cdot10$ |
| 63.1152.33-63.cp.3.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cq.1.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cr.4.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cs.2.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.ct.2.2 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cu.4.2 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cv.2.2 | $63$ | $3$ | $3$ | $33$ | $4$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cw.4.2 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cx.4.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cy.3.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.cz.4.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.da.3.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.33-63.db.1.3 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.dc.1.1 | $63$ | $3$ | $3$ | $33$ | $2$ | $6^{2}\cdot12$ |
| 63.1152.33-63.dd.3.4 | $63$ | $3$ | $3$ | $33$ | $8$ | $3^{4}\cdot12$ |
| 63.1152.37-63.a.2.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $1^{6}\cdot2^{5}\cdot4^{3}$ |
| 63.1152.37-63.bl.3.4 | $63$ | $3$ | $3$ | $37$ | $8$ | $1^{2}\cdot2\cdot3^{4}\cdot12$ |
| 63.1152.37-63.bm.3.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.bn.3.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.bo.2.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.bp.2.2 | $63$ | $3$ | $3$ | $37$ | $4$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.bq.2.4 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.br.1.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.bs.1.2 | $63$ | $3$ | $3$ | $37$ | $6$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.1152.37-63.bt.3.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot6^{2}\cdot12$ |
| 63.2688.93-63.a.2.1 | $63$ | $7$ | $7$ | $93$ | $20$ | $1^{18}\cdot2^{3}\cdot3^{4}\cdot4^{3}\cdot8\cdot12\cdot16$ |