Invariants
| Level: | $50$ | $\SL_2$-level: | $50$ | Newform level: | $250$ | ||
| Index: | $1800$ | $\PSL_2$-index: | $900$ | ||||
| Genus: | $56 = 1 + \frac{ 900 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (of which $4$ are rational) | Cusp widths | $5^{10}\cdot10^{10}\cdot25^{10}\cdot50^{10}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 15$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 15$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 50.1800.56.5 |
Level structure
| $\GL_2(\Z/50\Z)$-generators: | $\begin{bmatrix}1&16\\0&23\end{bmatrix}$, $\begin{bmatrix}9&7\\0&13\end{bmatrix}$ |
| $\GL_2(\Z/50\Z)$-subgroup: | $D_5.C_{10}^2$ |
| Contains $-I$: | no $\quad$ (see 50.900.56.a.2 for the level structure with $-I$) |
| Cyclic 50-isogeny field degree: | $1$ |
| Cyclic 50-torsion field degree: | $10$ |
| Full 50-torsion field degree: | $1000$ |
Jacobian
| Conductor: | $2^{24}\cdot5^{160}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{10}\cdot4^{8}$ |
| Newforms: | 50.2.a.a$^{2}$, 50.2.a.b$^{2}$, 50.2.b.a$^{2}$, 125.2.a.a$^{2}$, 125.2.a.b$^{2}$, 125.2.a.c$^{2}$, 125.2.b.a$^{2}$, 125.2.b.b$^{2}$, 250.2.a.a, 250.2.a.b, 250.2.a.c, 250.2.a.d, 250.2.b.a, 250.2.b.b |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|---|
| 10.360.4-10.a.1.3 | $10$ | $5$ | $5$ | $4$ | $0$ | $1^{2}\cdot2^{9}\cdot4^{8}$ |
| 50.360.4-50.a.2.3 | $50$ | $5$ | $5$ | $4$ | $0$ | $1^{2}\cdot2^{9}\cdot4^{8}$ |
| 50.600.16-25.a.1.2 | $50$ | $3$ | $3$ | $16$ | $2$ | $1^{4}\cdot2^{8}\cdot4^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 50.3600.121-50.a.2.2 | $50$ | $2$ | $2$ | $121$ | $12$ | $1^{9}\cdot2^{12}\cdot4^{8}$ |
| 50.3600.121-50.b.2.1 | $50$ | $2$ | $2$ | $121$ | $12$ | $1^{9}\cdot2^{12}\cdot4^{8}$ |
| 50.9000.276-50.a.1.3 | $50$ | $5$ | $5$ | $276$ | $6$ | $4^{9}\cdot8^{17}\cdot16^{3}$ |
| 50.9000.276-50.b.2.4 | $50$ | $5$ | $5$ | $276$ | $6$ | $4^{9}\cdot8^{17}\cdot16^{3}$ |
| 50.9000.276-50.c.1.3 | $50$ | $5$ | $5$ | $276$ | $6$ | $4^{9}\cdot8^{17}\cdot16^{3}$ |
| 50.9000.276-50.d.2.4 | $50$ | $5$ | $5$ | $276$ | $6$ | $4^{9}\cdot8^{17}\cdot16^{3}$ |
| 50.9000.276-50.e.1.3 | $50$ | $5$ | $5$ | $276$ | $54$ | $2^{18}\cdot4^{16}\cdot8^{15}$ |
| 50.9000.316-50.e.1.2 | $50$ | $5$ | $5$ | $316$ | $54$ | $1^{2}\cdot2^{21}\cdot4^{22}\cdot8^{12}\cdot16^{2}$ |