Invariants
| Level: | $50$ | $\SL_2$-level: | $50$ | Newform level: | $100$ | ||
| Index: | $720$ | $\PSL_2$-index: | $360$ | ||||
| Genus: | $13 = 1 + \frac{ 360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (of which $2$ are rational) | Cusp widths | $2^{30}\cdot50^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 50E13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 50.720.13.3 |
Level structure
| $\GL_2(\Z/50\Z)$-generators: | $\begin{bmatrix}21&8\\0&11\end{bmatrix}$, $\begin{bmatrix}39&7\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 50.360.13.b.1 for the level structure with $-I$) |
| Cyclic 50-isogeny field degree: | $1$ |
| Cyclic 50-torsion field degree: | $10$ |
| Full 50-torsion field degree: | $2500$ |
Jacobian
| Conductor: | $2^{18}\cdot5^{25}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{3}$ |
| Newforms: | 20.2.a.a, 50.2.a.a$^{2}$, 50.2.a.b$^{2}$, 50.2.b.a$^{2}$, 100.2.a.a$^{2}$, 100.2.c.a |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.144.1-10.b.1.1 | $10$ | $5$ | $5$ | $1$ | $0$ | $1^{6}\cdot2^{3}$ |
| 50.240.5-50.b.1.1 | $50$ | $3$ | $3$ | $5$ | $0$ | $1^{4}\cdot2^{2}$ |
| 50.360.4-50.a.2.1 | $50$ | $2$ | $2$ | $4$ | $0$ | $1^{5}\cdot2^{2}$ |
| 50.360.4-50.a.2.3 | $50$ | $2$ | $2$ | $4$ | $0$ | $1^{5}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 50.3600.109-50.m.1.2 | $50$ | $5$ | $5$ | $109$ | $0$ | $4^{5}\cdot8^{8}\cdot12$ |
| 50.3600.109-50.o.2.1 | $50$ | $5$ | $5$ | $109$ | $0$ | $4^{5}\cdot8^{8}\cdot12$ |
| 50.3600.109-50.q.2.1 | $50$ | $5$ | $5$ | $109$ | $0$ | $4^{5}\cdot8^{8}\cdot12$ |
| 50.3600.109-50.s.2.1 | $50$ | $5$ | $5$ | $109$ | $0$ | $4^{5}\cdot8^{8}\cdot12$ |
| 50.3600.109-50.u.2.2 | $50$ | $5$ | $5$ | $109$ | $24$ | $2^{10}\cdot4^{4}\cdot6^{2}\cdot8^{6}$ |
| 50.3600.121-50.b.2.1 | $50$ | $5$ | $5$ | $121$ | $12$ | $1^{6}\cdot2^{19}\cdot4^{16}$ |