Invariants
| Level: | $90$ | $\SL_2$-level: | $90$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{3}\cdot10^{3}\cdot18\cdot90$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 9$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 90F9 |
Level structure
| $\GL_2(\Z/90\Z)$-generators: | $\begin{bmatrix}33&38\\10&41\end{bmatrix}$, $\begin{bmatrix}44&13\\71&51\end{bmatrix}$, $\begin{bmatrix}69&82\\55&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 90.144.9.c.1 for the level structure with $-I$) |
| Cyclic 90-isogeny field degree: | $3$ |
| Cyclic 90-torsion field degree: | $72$ |
| Full 90-torsion field degree: | $38880$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 18.48.0-18.c.1.1 | $18$ | $6$ | $6$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 18.48.0-18.c.1.1 | $18$ | $6$ | $6$ | $0$ | $0$ |
| 30.96.3-30.b.1.6 | $30$ | $3$ | $3$ | $3$ | $0$ |