Invariants
| Level: | $50$ | $\SL_2$-level: | $50$ | Newform level: | $1250$ | ||
| Index: | $1800$ | $\PSL_2$-index: | $900$ | ||||
| Genus: | $48 = 1 + \frac{ 900 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 56 }{2}$ | ||||||
| Cusps: | $56$ (none of which are rational) | Cusp widths | $1^{10}\cdot2^{10}\cdot5^{8}\cdot10^{8}\cdot25^{10}\cdot50^{10}$ | Cusp orbits | $4^{4}\cdot5^{4}\cdot10^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $12$ | ||||||
| $\Q$-gonality: | $9 \le \gamma \le 15$ | ||||||
| $\overline{\Q}$-gonality: | $9 \le \gamma \le 15$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 50.1800.48.43 |
Level structure
| $\GL_2(\Z/50\Z)$-generators: | $\begin{bmatrix}11&6\\0&27\end{bmatrix}$, $\begin{bmatrix}19&43\\0&33\end{bmatrix}$ |
| $\GL_2(\Z/50\Z)$-subgroup: | $C_{50}:C_{20}$ |
| Contains $-I$: | no $\quad$ (see 50.900.48.k.1 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: | $2^{24}\cdot5^{184}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2^{7}\cdot4^{2}\cdot8^{3}$ |
| Newforms: | 50.2.a.a, 50.2.a.b, 50.2.b.a, 625.2.a.b$^{4}$, 625.2.b.c$^{2}$, 1250.2.a.a$^{2}$, 1250.2.a.l$^{2}$, 1250.2.b.c |
Rational points
This modular curve has no $\Q_p$ points for $p=3,13,17,23,37,53,67,113$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 50.360.4-50.a.2.3 | $50$ | $5$ | $5$ | $4$ | $0$ | $2^{6}\cdot4^{2}\cdot8^{3}$ |
| 50.600.12-25.k.1.2 | $50$ | $3$ | $3$ | $12$ | $4$ | $1^{2}\cdot2^{5}\cdot4^{2}\cdot8^{2}$ |
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.k.2.2 | $50$ | $2$ | $2$ | $109$ | $20$ | $1^{5}\cdot2^{6}\cdot4^{2}\cdot6^{2}\cdot8^{3}$ |
| 50.3600.109-50.u.2.2 | $50$ | $2$ | $2$ | $109$ | $24$ | $1^{5}\cdot2^{6}\cdot4^{2}\cdot6^{2}\cdot8^{3}$ |
| 50.9000.276-50.e.1.3 | $50$ | $5$ | $5$ | $276$ | $54$ | $1^{2}\cdot2^{21}\cdot4^{22}\cdot8^{12}$ |