Invariants
| Level: | $50$ | $\SL_2$-level: | $50$ | Newform level: | $2500$ | ||
| Index: | $2250$ | $\PSL_2$-index: | $2250$ | ||||
| Genus: | $161 = 1 + \frac{ 2250 }{12} - \frac{ 20 }{4} - \frac{ 0 }{3} - \frac{ 45 }{2}$ | ||||||
| Cusps: | $45$ (of which $3$ are rational) | Cusp widths | $50^{45}$ | Cusp orbits | $1^{3}\cdot4^{3}\cdot10\cdot20$ | ||
| Elliptic points: | $20$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $72$ | ||||||
| $\Q$-gonality: | $39 \le \gamma \le 75$ | ||||||
| $\overline{\Q}$-gonality: | $39 \le \gamma \le 75$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 50.2250.161.1 |
Level structure
| $\GL_2(\Z/50\Z)$-generators: | $\begin{bmatrix}0&1\\27&0\end{bmatrix}$, $\begin{bmatrix}0&3\\37&0\end{bmatrix}$ |
| $\GL_2(\Z/50\Z)$-subgroup: | $C_{20}\wr C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 50-isogeny field degree: | $2$ |
| Cyclic 50-torsion field degree: | $40$ |
| Full 50-torsion field degree: | $800$ |
Jacobian
| Conductor: | $2^{120}\cdot5^{576}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{28}\cdot4^{12}\cdot8^{6}$ |
| Newforms: | 20.2.a.a$^{2}$, 50.2.a.a$^{3}$, 50.2.a.b$^{3}$, 100.2.a.a, 125.2.a.a$^{3}$, 125.2.a.b$^{3}$, 125.2.a.c$^{3}$, 250.2.a.a$^{2}$, 250.2.a.b$^{2}$, 250.2.a.c$^{2}$, 250.2.a.d$^{2}$, 500.2.a.a, 500.2.a.b, 500.2.a.c, 625.2.a.a$^{2}$, 625.2.a.b$^{2}$, 625.2.a.c$^{2}$, 625.2.a.d, 625.2.a.e$^{2}$, 625.2.a.f, 625.2.a.g, 1250.2.a.a, 1250.2.a.b, 1250.2.a.c, 1250.2.a.d, 1250.2.a.e, 1250.2.a.f, 1250.2.a.g, 1250.2.a.h, 1250.2.a.i, 1250.2.a.j, 1250.2.a.k, 1250.2.a.l, 2500.2.a.b, 2500.2.a.e, 2500.2.a.f |
Rational points
This modular curve has 3 rational cusps and 1 rational CM point, but no other known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(10)$ | $10$ | $25$ | $25$ | $3$ | $0$ | $1^{6}\cdot2^{28}\cdot4^{12}\cdot8^{6}$ |
| 50.750.51.c.1 | $50$ | $3$ | $3$ | $51$ | $36$ | $1^{6}\cdot2^{20}\cdot4^{10}\cdot8^{3}$ |
| 50.1125.77.a.1 | $50$ | $2$ | $2$ | $77$ | $42$ | $1^{6}\cdot2^{11}\cdot4^{6}\cdot8^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(50)$ | $50$ | $2$ | $2$ | $331$ | $112$ | $1^{10}\cdot2^{26}\cdot4^{12}\cdot6^{2}\cdot8^{6}$ |
| 50.4500.331.f.1 | $50$ | $2$ | $2$ | $331$ | $132$ | $1^{10}\cdot2^{26}\cdot4^{12}\cdot6^{2}\cdot8^{6}$ |
| 50.11250.821.b.1 | $50$ | $5$ | $5$ | $821$ | $322$ | $1^{12}\cdot2^{91}\cdot4^{50}\cdot6^{7}\cdot8^{28}$ |