Invariants
| Level: | $99$ | $\SL_2$-level: | $9$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{3}\cdot9$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 9B0 |
Level structure
| $\GL_2(\Z/99\Z)$-generators: | $\begin{bmatrix}13&2\\10&48\end{bmatrix}$, $\begin{bmatrix}50&45\\56&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.12.0.a.1 for the level structure with $-I$) |
| Cyclic 99-isogeny field degree: | $12$ |
| Cyclic 99-torsion field degree: | $720$ |
| Full 99-torsion field degree: | $2138400$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 3100 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{15}(x^{3}-24y^{3})^{3}}{y^{9}x^{12}(x-3y)(x^{2}+3xy+9y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 33.8.0-3.a.1.1 | $33$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 99.72.0-9.d.1.2 | $99$ | $3$ | $3$ | $0$ |
| 99.72.0-9.d.2.1 | $99$ | $3$ | $3$ | $0$ |
| 99.72.0-9.e.1.1 | $99$ | $3$ | $3$ | $0$ |
| 99.72.1-9.a.1.1 | $99$ | $3$ | $3$ | $1$ |
| 99.288.9-99.a.1.2 | $99$ | $12$ | $12$ | $9$ |
| 198.48.1-18.a.1.2 | $198$ | $2$ | $2$ | $1$ |
| 198.48.1-198.a.1.2 | $198$ | $2$ | $2$ | $1$ |
| 198.48.1-18.b.1.2 | $198$ | $2$ | $2$ | $1$ |
| 198.48.1-198.b.1.2 | $198$ | $2$ | $2$ | $1$ |
| 198.72.0-18.a.1.3 | $198$ | $3$ | $3$ | $0$ |
| 297.72.0-27.a.1.2 | $297$ | $3$ | $3$ | $0$ |
| 297.72.1-27.a.1.2 | $297$ | $3$ | $3$ | $1$ |
| 297.72.2-27.a.1.2 | $297$ | $3$ | $3$ | $2$ |