Invariants
| Level: | $200$ | $\SL_2$-level: | $50$ | Newform level: | $1$ | ||
| Index: | $120$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $2^{10}\cdot50^{2}$ | Cusp orbits | $1^{2}\cdot2\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 50F5 |
Level structure
| $\GL_2(\Z/200\Z)$-generators: | $\begin{bmatrix}0&29\\81&127\end{bmatrix}$, $\begin{bmatrix}19&103\\90&41\end{bmatrix}$, $\begin{bmatrix}121&0\\165&111\end{bmatrix}$, $\begin{bmatrix}149&87\\135&12\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 200.240.5-200.b.2.1, 200.240.5-200.b.2.2, 200.240.5-200.b.2.3, 200.240.5-200.b.2.4, 200.240.5-200.b.2.5, 200.240.5-200.b.2.6, 200.240.5-200.b.2.7, 200.240.5-200.b.2.8 |
| Cyclic 200-isogeny field degree: | $12$ |
| Cyclic 200-torsion field degree: | $960$ |
| Full 200-torsion field degree: | $3840000$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.2.0.a.1 | $8$ | $60$ | $60$ | $0$ | $0$ |
| 25.60.0.a.2 | $25$ | $2$ | $2$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 25.60.0.a.2 | $25$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.1.bx.2 | $40$ | $5$ | $5$ | $1$ | $1$ |
| 200.60.2.b.1 | $200$ | $2$ | $2$ | $2$ | $?$ |
| 200.60.3.a.1 | $200$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 200.360.13.c.2 | $200$ | $3$ | $3$ | $13$ |