Invariants
| Level: | $204$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E2 |
Level structure
| $\GL_2(\Z/204\Z)$-generators: | $\begin{bmatrix}35&100\\10&175\end{bmatrix}$, $\begin{bmatrix}85&200\\194&35\end{bmatrix}$, $\begin{bmatrix}97&102\\42&5\end{bmatrix}$, $\begin{bmatrix}151&188\\160&67\end{bmatrix}$, $\begin{bmatrix}193&186\\0&101\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 204.36.2.e.1 for the level structure with $-I$) |
| Cyclic 204-isogeny field degree: | $144$ |
| Cyclic 204-torsion field degree: | $4608$ |
| Full 204-torsion field degree: | $5013504$ |
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 |
|---|---|---|---|---|---|
| 12.36.1-6.a.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 204.36.1-6.a.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 204.144.3-204.f.1.3 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.g.1.3 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.n.1.8 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.o.1.2 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.r.1.7 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.s.1.2 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.v.1.6 | $204$ | $2$ | $2$ | $3$ |
| 204.144.3-204.w.1.7 | $204$ | $2$ | $2$ | $3$ |
| 204.144.4-204.k.1.3 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.m.1.4 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.m.1.10 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.q.1.3 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.q.1.15 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.t.1.3 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.t.1.12 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.u.1.7 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.u.1.11 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.x.1.3 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.x.1.12 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.bg.1.3 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.bg.1.12 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.bj.1.2 | $204$ | $2$ | $2$ | $4$ |
| 204.144.4-204.bj.1.12 | $204$ | $2$ | $2$ | $4$ |