Invariants
| Level: | $204$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}9&142\\170&109\end{bmatrix}$, $\begin{bmatrix}15&166\\140&1\end{bmatrix}$, $\begin{bmatrix}32&111\\47&88\end{bmatrix}$, $\begin{bmatrix}52&105\\33&112\end{bmatrix}$, $\begin{bmatrix}141&62\\130&173\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 204-isogeny field degree: | $18$ |
| Cyclic 204-torsion field degree: | $1152$ |
| Full 204-torsion field degree: | $7520256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $8$ | $4$ | $0$ | $0$ |
| 51.8.0-3.a.1.1 | $51$ | $6$ | $6$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 102.24.0-6.a.1.1 | $102$ | $2$ | $2$ | $0$ | $?$ |
| 204.24.0-6.a.1.9 | $204$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 204.96.0-12.c.1.8 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-12.c.2.7 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-12.c.3.4 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-12.c.4.3 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-204.c.1.15 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-204.c.2.14 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-204.c.3.13 | $204$ | $2$ | $2$ | $0$ |
| 204.96.0-204.c.4.10 | $204$ | $2$ | $2$ | $0$ |
| 204.96.1-12.b.1.12 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-12.h.1.10 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-12.k.1.6 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.k.1.9 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-12.l.1.6 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.l.1.6 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.o.1.11 | $204$ | $2$ | $2$ | $1$ |
| 204.96.1-204.p.1.6 | $204$ | $2$ | $2$ | $1$ |
| 204.144.1-12.f.1.1 | $204$ | $3$ | $3$ | $1$ |