Invariants
| Level: | $45$ | $\SL_2$-level: | $9$ | ||||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $3$ are rational) | Cusp widths | $1^{3}\cdot3^{2}\cdot9^{3}$ | Cusp orbits | $1^{3}\cdot2\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 9I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 45.72.0.5 |
Level structure
| $\GL_2(\Z/45\Z)$-generators: | $\begin{bmatrix}17&0\\5&13\end{bmatrix}$, $\begin{bmatrix}24&7\\29&10\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 9.36.0.d.1 for the level structure with $-I$) |
| Cyclic 45-isogeny field degree: | $6$ |
| Cyclic 45-torsion field degree: | $48$ |
| 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 20 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^9}\cdot\frac{x^{36}(x^{3}+6x^{2}y-24y^{3})^{3}(x^{9}+18x^{8}y+108x^{7}y^{2}-48x^{6}y^{3}-4320x^{5}y^{4}-27936x^{4}y^{5}-95040x^{3}y^{6}-192384x^{2}y^{7}-221184xy^{8}-112128y^{9})^{3}}{y^{9}x^{36}(x+2y)^{9}(x+4y)^{9}(x^{2}+6xy+12y^{2})^{3}(x^{3}-36xy^{2}-72y^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 45.24.0-9.a.1.2 | $45$ | $3$ | $3$ | $0$ | $0$ |
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.2.1 | $45$ | $3$ | $3$ | $1$ |
| 45.360.12-45.d.1.3 | $45$ | $5$ | $5$ | $12$ |
| 45.432.11-45.d.2.6 | $45$ | $6$ | $6$ | $11$ |
| 45.720.23-45.d.2.1 | $45$ | $10$ | $10$ | $23$ |
| 90.144.3-18.g.1.2 | $90$ | $2$ | $2$ | $3$ |
| 90.144.3-18.k.1.2 | $90$ | $2$ | $2$ | $3$ |
| 90.144.3-90.k.1.4 | $90$ | $2$ | $2$ | $3$ |
| 90.144.3-90.l.1.4 | $90$ | $2$ | $2$ | $3$ |
| 90.216.2-18.d.1.4 | $90$ | $3$ | $3$ | $2$ |
| 135.216.1-27.a.1.1 | $135$ | $3$ | $3$ | $1$ |
| 135.216.4-27.b.1.1 | $135$ | $3$ | $3$ | $4$ |
| 135.216.4-27.d.1.1 | $135$ | $3$ | $3$ | $4$ |
| 180.144.3-36.i.1.3 | $180$ | $2$ | $2$ | $3$ |
| 180.144.3-36.l.1.2 | $180$ | $2$ | $2$ | $3$ |
| 180.144.3-180.ba.1.8 | $180$ | $2$ | $2$ | $3$ |
| 180.144.3-180.bb.1.8 | $180$ | $2$ | $2$ | $3$ |
| 180.288.9-36.cp.1.8 | $180$ | $4$ | $4$ | $9$ |