Invariants
| Level: | $200$ | $\SL_2$-level: | $25$ | ||||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $0 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $1^{10}\cdot25^{2}$ | Cusp orbits | $1^{2}\cdot2\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 25B0 |
Level structure
| $\GL_2(\Z/200\Z)$-generators: | $\begin{bmatrix}88&7\\163&19\end{bmatrix}$, $\begin{bmatrix}139&191\\22&185\end{bmatrix}$, $\begin{bmatrix}141&10\\28&49\end{bmatrix}$, $\begin{bmatrix}143&105\\103&6\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 25.60.0.a.2 for the level structure with $-I$) |
| Cyclic 200-isogeny field degree: | $12$ |
| Cyclic 200-torsion field degree: | $960$ |
| Full 200-torsion field degree: | $3840000$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 60 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(x-y)^{60}(1844144x^{20}-13905320x^{19}y+48979760x^{18}y^{2}-105184440x^{17}y^{3}+148393605x^{16}y^{4}-131615424x^{15}y^{5}+44718840x^{14}y^{6}+64400580x^{13}y^{7}-130561620x^{12}y^{8}+128577640x^{11}y^{9}-83857386x^{10}y^{10}+36648140x^{9}y^{11}-8423370x^{8}y^{12}-1656420x^{7}y^{13}+2723340x^{6}y^{14}-1521924x^{5}y^{15}+567105x^{4}y^{16}-152940x^{3}y^{17}+29510x^{2}y^{18}-3820xy^{19}+269y^{20})^{3}}{(x-y)^{60}(x+y)(3x-2y)(x^{2}-3xy+y^{2})^{25}(11x^{4}-31x^{3}y+41x^{2}y^{2}-31xy^{3}+11y^{4})(41x^{4}-51x^{3}y+26x^{2}y^{2}-6xy^{3}+y^{4})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-5.a.2.1 | $40$ | $5$ | $5$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 200.240.5-50.a.2.1 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-50.b.2.2 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-100.b.2.5 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-200.b.2.8 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-100.e.2.1 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-200.h.2.3 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-200.n.2.3 | $200$ | $2$ | $2$ | $5$ |
| 200.240.5-200.t.2.3 | $200$ | $2$ | $2$ | $5$ |
| 200.360.4-50.a.1.14 | $200$ | $3$ | $3$ | $4$ |
| 200.480.15-100.i.2.1 | $200$ | $4$ | $4$ | $15$ |