Invariants
| Level: | $63$ | $\SL_2$-level: | $63$ | Newform level: | $189$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| 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 $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 63C13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 63.384.13.14 |
Level structure
| $\GL_2(\Z/63\Z)$-generators: | $\begin{bmatrix}14&12\\23&10\end{bmatrix}$, $\begin{bmatrix}19&45\\51&62\end{bmatrix}$, $\begin{bmatrix}49&9\\9&35\end{bmatrix}$, $\begin{bmatrix}52&36\\15&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 63.192.13.a.1 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: | $3^{37}\cdot7^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{3}\cdot4$ |
| Newforms: | 21.2.a.a, 27.2.a.a$^{2}$, 189.2.a.e, 189.2.a.f, 189.2.c.a, 189.2.c.b |
Models
Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ x d - a b $ |
| $=$ | $x t - y a - s a$ | |
| $=$ | $x d + t u + r d$ | |
| $=$ | $y d - t b + s d$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y^{4} - 2 x^{6} y^{3} z + 3 x^{6} y^{2} z^{2} - 2 x^{6} y z^{3} + x^{6} z^{4} + 5 x^{3} y^{6} z + \cdots + y z^{9} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:-1:0:0:0:0:0:0:1:0:0:0:0)$, $(0:0:1:0:0:0:0:0:0:0:0:1:0)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 63.96.7.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y-z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -t$ |
| $\displaystyle T$ | $=$ | $\displaystyle x+y-z+v+r$ |
| $\displaystyle U$ | $=$ | $\displaystyle t-a+b$ |
| $\displaystyle V$ | $=$ | $\displaystyle -d$ |
Equation of the image curve:
| $0$ | $=$ | $ W^{2}-XV+ZV-TV $ |
| $=$ | $ W^{2}+WU-XV-YV $ | |
| $=$ | $ YW+ZW-WT-XU+ZU-TU $ | |
| $=$ | $ X^{2}-XY-XZ+YZ-Z^{2}+XT-YT+ZT $ | |
| $=$ | $ 2XW+2YW+XU+YU-V^{2} $ | |
| $=$ | $ 2XY+Y^{2}-2XZ-XT $ | |
| $=$ | $ 2X^{2}+XY+XZ-YZ+2XT+YT-WV $ | |
| $=$ | $ WU+U^{2}-3XV+4YV-3ZV $ | |
| $=$ | $ 3XW-4YW+3ZW-XU-YU $ | |
| $=$ | $ 3X^{2}+2XY-2Y^{2}+2XZ+4YZ-2XT-YT-UV $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 63.192.13.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle d$ |
| $\displaystyle Y$ | $=$ | $\displaystyle a$ |
| $\displaystyle Z$ | $=$ | $\displaystyle b$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}Y^{4}-2X^{6}Y^{3}Z+5X^{3}Y^{6}Z+Y^{9}Z+3X^{6}Y^{2}Z^{2}-15X^{3}Y^{5}Z^{2}-2Y^{8}Z^{2}-2X^{6}YZ^{3}+34X^{3}Y^{4}Z^{3}-8Y^{7}Z^{3}+X^{6}Z^{4}-34X^{3}Y^{3}Z^{4}+83Y^{6}Z^{4}+15X^{3}Y^{2}Z^{5}-148Y^{5}Z^{5}-5X^{3}YZ^{6}+83Y^{4}Z^{6}-8Y^{3}Z^{7}-2Y^{2}Z^{8}+YZ^{9} $ |
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 | $21$ | $3$ | $3$ | $1$ | $0$ | $1^{2}\cdot2^{3}\cdot4$ |
| 63.192.7-63.a.1.6 | $63$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 63.192.7-63.a.1.7 | $63$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 63.1152.37-63.a.2.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $1^{8}\cdot2^{2}\cdot4^{3}$ |
| 63.1152.37-63.b.1.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
| 63.1152.37-63.c.3.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.d.1.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.e.3.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.f.3.1 | $63$ | $3$ | $3$ | $37$ | $8$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.g.3.2 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.h.1.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.i.1.1 | $63$ | $3$ | $3$ | $37$ | $4$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.j.1.2 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{6}\cdot4^{3}$ |
| 63.1152.37-63.k.4.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.l.2.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.m.3.1 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.n.2.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.o.4.4 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.p.1.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.q.2.4 | $63$ | $3$ | $3$ | $37$ | $4$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.r.1.4 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.s.4.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.t.3.3 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.u.1.4 | $63$ | $3$ | $3$ | $37$ | $12$ | $1^{2}\cdot2\cdot5^{2}\cdot10$ |
| 63.1152.37-63.v.3.4 | $63$ | $3$ | $3$ | $37$ | $12$ | $1^{2}\cdot2\cdot5^{2}\cdot10$ |
| 63.1152.37-63.w.2.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.x.1.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.y.4.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.z.3.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.ba.1.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bb.2.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bc.3.1 | $63$ | $3$ | $3$ | $37$ | $2$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bd.4.1 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.be.2.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bf.1.2 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bg.4.4 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bh.3.4 | $63$ | $3$ | $3$ | $37$ | $0$ | $2^{2}\cdot10^{2}$ |
| 63.1152.37-63.bi.2.4 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.bj.3.3 | $63$ | $3$ | $3$ | $37$ | $0$ | $6^{2}\cdot12$ |
| 63.1152.37-63.bk.2.4 | $63$ | $3$ | $3$ | $37$ | $6$ | $3^{4}\cdot12$ |
| 63.2688.97-63.a.2.1 | $63$ | $7$ | $7$ | $97$ | $12$ | $1^{18}\cdot2^{11}\cdot4^{5}\cdot8\cdot16$ |