Invariants
| Level: | $204$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}17&166\\30&127\end{bmatrix}$, $\begin{bmatrix}50&199\\35&102\end{bmatrix}$, $\begin{bmatrix}63&200\\56&27\end{bmatrix}$, $\begin{bmatrix}95&30\\80&73\end{bmatrix}$, $\begin{bmatrix}199&44\\172&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 204-isogeny field degree: | $36$ |
| Cyclic 204-torsion field degree: | $2304$ |
| Full 204-torsion field degree: | $15040512$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 204.48.0-6.a.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-102.a.1.5 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-6.b.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-102.b.1.11 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-12.d.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-12.f.1.1 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-12.g.1.2 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-12.h.1.7 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-12.i.1.3 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-12.j.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.o.1.14 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.p.1.16 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.q.1.15 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.r.1.14 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.s.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.48.0-204.t.1.7 | $204$ | $2$ | $2$ | $0$ |
| 204.48.1-12.i.1.1 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-12.j.1.2 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-12.k.1.3 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-12.l.1.8 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.m.1.10 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.n.1.9 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.o.1.2 | $204$ | $2$ | $2$ | $1$ |
| 204.48.1-204.p.1.1 | $204$ | $2$ | $2$ | $1$ |
| 204.72.0-6.a.1.2 | $204$ | $3$ | $3$ | $0$ |
| 204.432.15-102.a.1.33 | $204$ | $18$ | $18$ | $15$ |