Invariants
| Level: | $50$ | $\SL_2$-level: | $50$ | Newform level: | $250$ | ||
| 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 | $5^{20}\cdot10^{20}\cdot25^{8}\cdot50^{8}$ | Cusp orbits | $2^{10}\cdot4^{4}\cdot5^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-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: | 50.1800.48.2 |
Level structure
| $\GL_2(\Z/50\Z)$-generators: | $\begin{bmatrix}23&37\\20&1\end{bmatrix}$, $\begin{bmatrix}29&43\\20&21\end{bmatrix}$ |
| $\GL_2(\Z/50\Z)$-subgroup: | $C_5\times C_{10}:F_5$ |
| Contains $-I$: | no $\quad$ (see 50.900.48.c.1 for the level structure with $-I$) |
| Cyclic 50-isogeny field degree: | $5$ |
| Cyclic 50-torsion field degree: | $20$ |
| Full 50-torsion field degree: | $1000$ |
Jacobian
| Conductor: | $2^{24}\cdot5^{120}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2\cdot4^{3}\cdot8^{4}$ |
| Newforms: | 25.2.d.a$^{2}$, 50.2.a.a, 50.2.a.b, 50.2.b.a, 50.2.d.a, 50.2.d.b, 125.2.e.b$^{2}$, 250.2.e.a |
Rational points
This modular curve has no $\Q_p$ points for $p=3,13,17,47,83,97$, and therefore no 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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $600$ | $300$ | $0$ | $0$ | full Jacobian |
| 25.600.12-25.c.1.2 | $25$ | $3$ | $3$ | $12$ | $0$ | $1^{2}\cdot2\cdot4^{2}\cdot8^{3}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.360.4-10.a.1.2 | $10$ | $5$ | $5$ | $4$ | $0$ | $4^{3}\cdot8^{4}$ |
| 25.600.12-25.c.1.2 | $25$ | $3$ | $3$ | $12$ | $0$ | $1^{2}\cdot2\cdot4^{2}\cdot8^{3}$ |
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.c.1.2 | $50$ | $2$ | $2$ | $109$ | $0$ | $1^{5}\cdot2^{2}\cdot4^{2}\cdot8^{4}\cdot12$ |
| 50.3600.109-50.p.1.2 | $50$ | $2$ | $2$ | $109$ | $0$ | $1^{5}\cdot2^{2}\cdot4^{2}\cdot8^{4}\cdot12$ |
| 50.9000.276-50.c.1.2 | $50$ | $5$ | $5$ | $276$ | $6$ | $1^{2}\cdot2^{9}\cdot4^{14}\cdot8^{13}\cdot16^{3}$ |
| 50.9000.276-50.c.2.2 | $50$ | $5$ | $5$ | $276$ | $6$ | $1^{2}\cdot2^{9}\cdot4^{14}\cdot8^{13}\cdot16^{3}$ |