Invariants
| Level: | $45$ | $\SL_2$-level: | $9$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 3 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $3$ are rational) | Cusp widths | $3^{3}\cdot9^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $3$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9J0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 45.72.0.16 |
Level structure
| $\GL_2(\Z/45\Z)$-generators: | $\begin{bmatrix}26&40\\0&4\end{bmatrix}$, $\begin{bmatrix}35&36\\32&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.36.0.f.1 for the level structure with $-I$) |
| Cyclic 45-isogeny field degree: | $18$ |
| Cyclic 45-torsion field degree: | $432$ |
| Full 45-torsion field degree: | $25920$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(x+2y)^{36}(x^{2}+xy+y^{2})^{3}(x^{3}-3x^{2}y-6xy^{2}-y^{3})(x^{9}-9x^{7}y^{2}-246x^{6}y^{3}-702x^{5}y^{4}-711x^{4}y^{5}-267x^{3}y^{6}-27x^{2}y^{7}-9xy^{8}-y^{9})^{3}}{y^{3}x^{3}(x+y)^{3}(x+2y)^{36}(x^{3}-3xy^{2}-y^{3})^{9}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 45.24.0-9.b.1.1 | $45$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 45.216.1-9.a.1.1 | $45$ | $3$ | $3$ | $1$ |
| 45.360.11-45.a.2.2 | $45$ | $5$ | $5$ | $11$ |
| 45.432.13-45.d.2.3 | $45$ | $6$ | $6$ | $13$ |
| 45.720.24-45.a.2.5 | $45$ | $10$ | $10$ | $24$ |
| 90.144.2-90.a.2.2 | $90$ | $2$ | $2$ | $2$ |
| 90.144.2-18.b.1.2 | $90$ | $2$ | $2$ | $2$ |
| 90.144.2-90.b.2.2 | $90$ | $2$ | $2$ | $2$ |
| 90.144.2-18.e.1.2 | $90$ | $2$ | $2$ | $2$ |
| 90.216.4-18.g.1.2 | $90$ | $3$ | $3$ | $4$ |
| 135.216.4-27.f.1.1 | $135$ | $3$ | $3$ | $4$ |
| 135.216.7-27.c.1.1 | $135$ | $3$ | $3$ | $7$ |
| 135.216.7-27.e.1.1 | $135$ | $3$ | $3$ | $7$ |
| 180.144.2-36.a.2.4 | $180$ | $2$ | $2$ | $2$ |
| 180.144.2-180.a.2.6 | $180$ | $2$ | $2$ | $2$ |
| 180.144.2-180.b.1.6 | $180$ | $2$ | $2$ | $2$ |
| 180.144.2-36.c.2.4 | $180$ | $2$ | $2$ | $2$ |
| 180.288.9-36.cr.2.8 | $180$ | $4$ | $4$ | $9$ |